Stabilization of Ambient Pressure Rocksalt Crystal Structure and High Critical Field Superconductivity in ReC via Mo and W Substitution
P. K. Meena S. Jangid R. K. Kushwaha P. Manna S. Sharma P. Mishra R. P. Singh




Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

[]rpsingh@iiserb.ac.in
Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India

§ ABSTRACT 
Transition-metal-based carbides (TMCs), renowned for their exceptional hardness, mechanical strength, and thermal properties, have recently emerged as promising candidates for topological superconductivity. In this study, we synthesized ReC in the NaCl structure at ambient pressure by substituting Mo or W at the Re-site. We investigated the superconducting properties of Re_1-xT_xC (where T = Mo, W) for x = 0.5 using magnetization, resistivity and specific heat measurements. These compounds display type-II, fully gapped, weakly coupled superconductivity with high critical fields, establishing them as new members of superconducting ultra-hard materials at ambient pressure and paving the way for superconducting device applications under extreme conditions.



Stabilization of Ambient Pressure Rocksalt Crystal Structure and High Critical Field Superconductivity in ReC via Mo and W Substitution
    R. P. Singh 0000-0003-2548-231X
    September 9, 2024
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Transition-metal carbides (TMCs) are a class of refractory, ultra-hard materials with a unique combination of complex covalent, ionic, and metallic bonding, resulting in exceptional mechanical and electronic properties and making TMCs promising candidates for advanced material applications. Similar to 2D materials such as graphene and transition-metal dichalcogenides <cit.>. Renowned for their superconductivity, high melting points, outstanding strength, corrosion resistance, and excellent thermal and electrical conductivity, TMCs have found applications in cutting tools, wear-resistant coatings, and catalysts. Their ultra-hardness provides a viable alternative to diamonds, which are often costly and difficult to synthesize <cit.>. Moreover, the properties of TMCs can be tailored by controlling their valence electron concentration (VEC)<cit.>. However, the synthesis of TMCs can be challenging due to the off-stoichiometric carbon content.

The superconductivity of these refractory materials makes TMCs suitable candidates for device applications in extreme conditions. Recently, rocksalt-structured superconducting TMCs have emerged as prime candidates for hosting topological superconductivity due to their non-trivial band topology <cit.>. Some of these carbides, such as XC (where X = Nb, Mo, Ta, V, and Cr) <cit.>, exhibit intriguing non-trivial features, including Fermi surface nesting, nodal line, and the semimetallic nature of Dirac <cit.>, along with a high superconducting transition temperature (T_C) compared to other reported topological superconductors. Isostructural ScS shows unconventional properties with time-reversal symmetry breaking (TRSB) <cit.>. Rocksalt-structured carbides like (Nb, Ta, Mo, W)C show a transition temperature (T_C) of up to 14 K <cit.>. Recent theoretical calculations also suggest the possibility of unconventional superconductivity in TMCs <cit.>.

Re-based carbides are particularly notable for their exceptional high-temperature resistance and superior mechanical properties, which surpass those of diamond under high-pressure conditions, making them highly desirable for industrial applications. Although the superconductivity and ultra-incompressibility of ReC have been studied theoretically, stabilizing the rocksalt structure of ReC under ambient conditions remains a challenge <cit.>.

In this paper, we report the stabilization of the NaCl-type structure at ambient pressure and the induction of superconductivity in non-superconducting ReC through the Mo and W substitution. We present a comprehensive study of the structural and superconducting properties of the compounds Re_1-xT_xC (where T = Mo and W) for x = 0.5. These compounds have been predicted to exhibit remarkable mechanical properties, including ultra-compressibility and superhardness, making them promising candidates for advanced applications <cit.>. The superconductivity of these compounds was examined by resistivity, magnetization, and specific heat measurements, revealing bulk type-II superconductivity with weakly coupled BCS characteristics.




Polycrystalline Re_1-xT_xC (T = Mo, W) was synthesized using an arc melter with high-purity (4N) Mo, W, Re, and C in stoichiometric ratios of 1-x:x:1. The materials were melted in an Ar atmosphere on a water-cooled copper hearth and remelted several times for homogeneity. The crystal structure and phase purity were analyzed by powder X-ray diffraction (XRD) with CuKα radiation. Magnetic properties were measured with a Quantum Design MPMS-3, while electrical resistivity and specific heat were measured using the four-probe method and the two-tau time relaxation method on a Quantum Design PPMS 9T system.

 


The XRD patterns of Re_1-xT_xC compounds (where T = Mo, W) with Mo: x = 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, 0.9) and W: x = 0.2, 0.5, 0.8) are presented in Fig1(a) and (e). It suggests a single-phase NaCl structure stabilized between x = 0.2 - 0.6 for Mo and x = 0.5 for the substitution of W. The structural refinement of the XRD patterns was performed using a Rietveld method using FULLPROF software <cit.>, a NaCl-type face-centered cubic crystal structure with space group Fm3̅m (no. 225) and the phase purity of the sample. The refined XRD patterns for Re_0.5T_0.5C (T = Mo, W) are shown in Fig1(b) and (f). The inset illustrates the NaCl-type structure, with mixed T and Re atoms at the 4a (0, 0, 0) site and carbon atoms at the 4b (0.5, 0.5, 0.5) site.
The lattice parameters, listed in Table <ref>, are consistent with published data for compositions x = 0.5 <cit.> and slightly smaller than those of binary carbides <cit.>. This reduction is attributed to the smaller atomic radius of Re. The observed instability with 20% W substitution compared to Mo substitution might be due to the inherently stable NaCl structure of MoC and the intrinsically unstable NaCl structure of WC under normal pressure. These findings emphasize the complex relationship between substitution levels and structural stability in these materials.

Fig1(c) and (g) show magnetization measurements in zero-field-cooled warming (ZFCW) and field-cooled cooling (FCC) modes, with a 1.0 mT applied field, confirmed superconductivity in both compounds. The superconducting transition temperatures (T_C) are listed in Table <ref>, for Re_1-xT_xC (T = Mo, W). The stronger diamagnetic response in FCC compared to ZFCW due to flux trapping confirms the type-II nature of these superconductors. The superconducting fractions exceed 100% due to the irregular sample shape and demagnetization effects <cit.>. The hysteresis magnetization loops for x = 0.5 compositions (insets in Fig1(c) and (g)) further support type-II superconductivity, with H_irr values of 1.6 and 1.8 T for Mo and W-based carbides, respectively.

Figs. <ref>(d) and (h) show the temperature-dependent AC electrical resistivity ρ(T) of Re_0.5T_0.5C (T = Mo, W) carbides, measured from 1.9 to 300 K in a zero magnetic field. Above the superconducting transition temperature (T_C), ρ(T) remains nearly constant, indicating poor metallic behavior. The residual resistivity ratio (RRR = ρ_300/ρ_10) for the Mo and W variants, 1.03(7) and 1.01(8), respectively, indicating high intrinsic disorder in the compounds. These values are similar to those reported for binary carbides and high-entropy alloys. The inset of Figs. <ref>(d) and (h) shows a drop in resistivity at T_C = 3.86(4) K for Re_0.5Mo_0.5C and 3.92(7) K for Re_0.5W_0.5C, confirming superconductivity. The observed T_C values are consistent with the magnetization measurement.


The resistivity data above T_C can be well understood by the Bloch-Gruneisen (BG) model with n = 3 expressed as <cit.>,

    ρ(T)= ρ_0 + A (T/θ_D)^3 ∫_0^θ_D/Tx^3/(e^x-1)(1-e^-x) dx

where ρ_0 is the residual resistivity, A is the degree of electron correlation in the substance <cit.> and θ_D is the Debye temperature. The fitted resistivity data from the BG model provides A = 2.27 μΩ.cm, θ_D = 283.06, ρ_0 = 142.64 μΩ.cm for Re_0.5Mo_0.5C and A = 2.32 μΩ.cm, θ_D = 287.06, ρ_0 = 307.12 μΩ.cm for Re_0.5W_0.5C. These values of θ_D are relatively close to those obtained from the heat capacity data (discussed later).





Low-field magnetization curves were recorded at various temperatures to determine the lower critical field, H_C1(0). The onset of vortex formation and the mixed state was identified as the deviation point from the Meissner line in the M-H curves (insets in Figs. <ref>(a) and (c)). The temperature-dependent H_C1 data was analyzed using the Ginzburg-Landau (GL) equation, expressed as: H_C1(T) = H_C1(0)[1-(T/T_C)^2] providing H_C1(0) values of 2.68(1) and 3.15(3) mT for Re_0.5T_0.5C (T = Mo, W), respectively.

To calculate the upper critical field, H_C2(0), the effect of the applied magnetic field on T_C was measured through magnetization and electrical resistivity (insets in Figs. <ref>(b) and (d)). The temperature-dependent H_C2 data was analyzed using the GL equation, expressed as: H_C2(T) = H_C2(0)[1-t^2/1+t^2], yielding H_C2(0) values of 3.60(3) and 3.92(2) T for Re_0.5Mo_0.5C, and 4.06(4) and 4.42(3) T for Re_0.5W_0.5C, from magnetization and resistivity measurements, respectively.

Temperature-dependent H_C1(0) and H_C2(0) plots, along with GL fits, are shown for both samples in Figs. <ref>(a) and (c) and Figs. <ref>(b) and (d), respectively. The obtained H_C2(0) and H_C1(0) values were used to calculate the superconducting characteristic length scales, coherence length λ_GL and penetration length ξ_GL <cit.>. The expressions for these parameters are given by:

    H_C2(0) = Φ_0/2πξ_GL^2.


    H_C1(0) = Φ_0/4πλ_GL^2( ln λ_GL/ξ_GL + 0.12).

where Φ_0 is the magnetic flux quantum. 
The obtained parameters λ_GL and ξ_GL for the Mo-variant are 5047(5) Å and 95.6(6) Å. For the W variant, they are 4592(7) Å and 90.1(8) Å. The derived GL parameters κ_GL (= λ_GL/ξ_GL) are 52.20(3) and 50.98(5) for Re_0.5Mo_0.5C and Re_0.5W_0.5C. These values exceed 1 /√(2), confirming these superconductors' type II nature. Furthermore, using H_C1(0), H_C2(0) and κ_GL, the thermodynamic critical field H_C was calculated using the relation: H_C^2 ln[k_GL(0)+0.08] = H_C1(0) H_C2(0) <cit.>, yielding values of 48.8(6) and 57.1(3) mT for the Mo and W-based carbides, respectively.


Superconductivity can be destroyed by orbital and Pauli paramagnetic limiting field effects. The orbital limit, H_C2^orb(0), is associated with the kinetic energy of the Cooper pair. The Werthamer-Helfand-Hohenberg (WHH) theory provides the expression <cit.>:

    H^orb_C2(0) = -α T_C.dH_C2(T)/dT|_T=T_C.

where α is the purity factor (0.69 for dirty, 0.73 for clean limits). The initial slope, -dH_C2(T)/dT|_T=T_C, was obtained from the H_C2-T phase diagram (Figs. <ref>(b) and (d)). The slope values obtained are 0.96(7) and 0.78(6) T / K for carbides doped with Mo and W, respectively. For dirty limit superconductors (α = 0.69), the calculated H_C2^orb(0) values are 2.54(7) and 2.06 (9) T for carbides based on Mo and W.

The Pauli paramagnetic limiting field, H_C2^P(0), arises from the Zeeman splitting and is given by: H_C2^P(0) = 1.86*T_C for weakly coupled BCS superconductors <cit.>. The estimated H_C2^P values are 7.01(2) and 7.03(1) T for Mo and W-based carbides. Since the upper critical field is lower than the Pauli limiting field, the orbital limiting field is responsible for breaking the Cooper pairs in both compounds.

To quantify spin paramagnetic effects, we calculated the Maki parameter, (α_m), using the expression α_m = √(2) H_C2^orb(0)/H_C2^P(0). The values of α_m are 0.51(2) and 0.41(4) for Mo and W variants, respectively.

The Ginzburg number G_i, a ratio of thermal energy to condensation energy related to the coherence volume, is given by:

    G_i = 1/2[k_Bμ_0τ T_C/4 πξ^3_GL(0) H^2_C(0)]^2,

where τ is an anisotropic ratio and assumes 1 for cubic structure. Using the values of T_C, ξ_GL(0) and H_C, we calculated G_i = 3.15(1) and 2.43(2) × 10^-6 for Mo and W-based carbides. These values are lower than those of high-T_C cuprate superconductors (10^-2) <cit.> but higher than those of low-T_C superconductors (10^-8) <cit.> suggesting that weak thermal fluctuations contribute to vortex unpinning in these carbides <cit.>.



Specific heat measurements were performed in zero magnetic field for Re_0.5T_0.5C (T = Mo, W), as shown in Figs. <ref>(a) and (b). A clear transition from the normal to the superconducting state was observed at T_C = 3.7(1) K for both Mo- and W-based carbides, consistent with magnetization and resistivity measurements. The Debye-Sommerfeld relation, C/T = γ_n + β_3T^2, where γ_n is the Sommerfeld coefficient and β_3 is the phononic coefficient β_3, used to analyze normal state behavior. By extrapolating the normal state behavior to low temperatures, we obtained γ_n and β_3 are 6.60(2) mJ/mol K^2 and 0.219(8) mJ/mol K^4 for Re_0.5Mo_0.5C and 6.18(3) mJ/mol K^2 and 0.209(3) mJ/mol K^4 for Re_0.5W_0.5C, respectively. β_3, was used to calculate the Debye temperature, θ_D using the relation θ_D = (12π^4 R N/5 β_3)^1/3, where N is the number of atoms per formula unit and R = 8.314 J mol^-1 K^-1 is a gas constant. The calculated values for θ_D are 328(2) and 333(7) K for Mo and W-based carbides, consistent with resistivity data. The Sommerfeld coefficient γ_n, is related to the density of states at the Fermi energy, D_C(E_F), as γ_n = (π^2 k_B^2/3) D_C(E_F). Here, k_B = 1.38 × 10^-23 J K^-1 is the Boltzmann constant. The calculated values of D_C(E_F) are 2.79(8) and 2.62(2) states eV^-1 f.u.^-1 for Re_0.5T_0.5C (T = Mo, W respectively). These values are higher than the topological isostructural rock salt carbides (Nb, Ta)C <cit.>. 

McMillan's theory <cit.> provides a formula to calculate the dimensionless electron-phonon coupling parameter λ_ep, which represents the strength of the attractive interaction between electrons and phonons. The expression is defined as:

    T_C = θ_D/1.45exp[1.04(1+λ_ep)/μ^* (1+0.62.λ_ep)-λ_ep];

where μ^* is the repulsive-screened Coulomb potential parameter (0.13 for intermetallic compounds). The estimated λ_ep values for the Mo and W-variants are 0.574(2) and 0.571(8), respectively, indicating weak-coupled superconductivity.

Specific heat measurements also provide insights into the superconducting gap symmetry. By subtracting the phononic terms from the total specific heat at zero fields, we calculated the electronic contribution C_el. The specific heat jump Δ(0)/γ_nT_C was found to be 1.30 (2) and 1.29 (4) for the Mo- and W-based carbide, slightly lower than the BCS value of 1.43. Low-temperature C_el was theoretically fitted by the α-model of the isotropic BCS weak coupling superconductor <cit.>, represented by the equation:

    S/γ_n T_C= -6/π^2(Δ(0)/k_B T_C) ∫_0^∞[ fln(f)+(1-f)ln(1-f)] dy

where f(ξ) = (exp(E(ξ)/k_BT)+1)^-1 is the Fermi function, and y = ξ/Δ(0), with E(ξ) = √(ξ^2+Δ^2(t)), which is the energy of normal electrons relative to the Fermi energy, where t = T/T_C is normalized temperature. The normalized temperature-dependent gap function is written as Δ(t)=tanh[1.82((1.018(1/t))-1)^0.51]. Using the relation between entropy S and C_el, given as C_el = tdS/dt, we calculated the electronic specific heat. The superconducting gap (α = Δ(0)/k_BT_C), was found to be 1.74(3) and 1.73(5) for carbides based on Mo and W, which is close to the BCS value for isotropic, weakly coupled superconductors, suggesting phonon-mediated superconductivity in these ultra-hard materials. The α-model of the isotropic BCS superconductor's best fitting of Figs. <ref>(a) and (b) provides us with this ratio. Data points up to T_C/10, and microscopic measurements are required to determine the exact pairing mechanism.

To study the electronic properties of these Re-based carbides, we employed a set of equations considering a spherical Fermi surface. These equations allow us to calculate the mean free path, l_e, and the effective mass, m^* <cit.>. The equations are as follows:

    γ_n = (π/3)^2/3k_B^2m^*V_f.u.n^1/3/ħ^2N_A


    l_e = 3π^2ħ^3/e^2ρ_0m^*2v_F^2, n = 1/3π^2(m^*v_F/ħ)^3

where k_B, V_f.u., and N_A are the Boltzmann constant, the volume of a formula unit, and the Avogadro number, respectively. For a dirty limit superconductor, the effective magnetic penetration depth λ_GL(0) can be expressed in terms of the London penetration depth λ_L(0) as: 

    λ_GL(0) = 
    λ_L(1+ξ_0/l_e)^1/2, λ_L =
    (m^*/μ_0n e^2)^1/2


    ξ_GL(0)/ξ_0 = π/2√(3)(1+ξ_0/l_e)^-1/2




The high value of ξ_0/l_e suggests that both Re-based carbides have a dirty limit superconductivity. The calculated electronic parameters are summarized in tbl: parameters. 

The Uemura plot helps categorize superconductors as conventional or unconventional based on Fermi temperature and superconducting transition temperature <cit.>. We calculated the Fermi temperature for these carbides using the equation <cit.>:

    k_BT_F = ħ^2/2(3π^2)^2/3n^2/3/m^*.


The values obtained for carbides based on Mo and W are T_F are 3.38(6) and 3.61(4) × 10^4 K. The corresponding T_C/T_F ratios, 0.00011 and 0.00010, fall outside the unconventional band range (0.01 < T_C/T_F < 0.05) in the Uemura plot, indicating that these carbides are conventional superconductors.







In conclusion, we successfully synthesized ReC, a rocksalt-type crystal structure, at ambient pressure through Mo and W doping. Magnetization, electrical resistivity, and specific heat measurements revealed bulk type-II superconductivity in Re_0.5T_0.5C (T = Mo, W) with T_C ≈ 3.8 K. These compounds exhibit face-centered-cubic NaCl-type structures and weak-coupling s-wave superconductivity. The superconducting properties of Re_0.5T_0.5C are comparable to those observed in Mo and W-based borides with the same valence electron count <cit.>. Further exploration of their mechanical properties is warranted. The combination of potential non-trivial band topology, high spin-orbit coupling elements, enhanced hardness, and high critical field superconductivity in Re_0.5T_0.5C provides valuable insights into the relationships between these properties and their potential for practical applications in extreme conditions. Given the time-reversal symmetry breaking observed in ScS <cit.>, these Re-based carbides may offer insights into the frequent occurrence of unconventional superconducting ground states in Re-based superconductors. Additionally, this study paves the way for the development of ultra-hard superconducting materials by exploring the formation of high-entropy carbides through the addition of suitable elements.



Acknowledgments: P.K.M. acknowledges the financial support provided by the CSIR, Government of India, through the SRF Fellowship (Award No: 09/1020(0174)/2019-EMR-I). R.P.S. is thankful to the SERB, Government of India, for the Core Research Grant (CRG/2023/000817).

50
TMCs1 E. Wuchina, E. Opila, M. Opeka, W. Fahrenholtz, and I. Talmy, Electrochem. Soc. Interface 16, 30 (2007).
TMCs2 L. E. Toth, Transition Metal Carbides and Nitrides (Academic, New York, 1971).
Carbides1 V. A. Gubanov, A. L. Ivanovsky, and V. P. Zhukov (Cambridge, New York: Cambridge University Press) pp. 18, 57 (1994).
Carbides2 X. Y. Yang, Y. Lu, F. W. Zheng, and P. Zhang, Chin. Phys. B 24, 116301 (2015).
2D-TMCs T. Qin, Z. Wang, Y. Wang, F. Besenbacher, M. Otyepka, and M. Dong, Nano-Micro Letters 13, 183 (2021).
Legar J. M. Leger, J. Low Temp. Phys. 14, 297 (1974).



TopologicalSC X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
Majorana-fermion A. Y. Kitaev, Phys. Usp. 44, 131 (2001).
TSC C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80, 1083 (2008).
Nb/TaC T. Shang, J. Z. Zhao, D. J. Gawryluk, M. Shi, M. Medarde, E. Pomjakushina, and T. Shiroka, Phys. Rev. B 101, 214518 (2020).
V/CrC R. Zhan and X. Luo, J. Appl. Phys. 125, 053903 (2019).
MoC1 C. I. Sathish, Y. Shirako, Y. Tsujimoto, H. L. Feng, Y. Sun, M. Akaogi, and K. Yamaura, Solid State Commun. 177, 33 (2014).
TaC-hexagonal1 D. Y. Yan, M. Yang, C. X. Wang, P. B. Song, C. J. Yi, Y. G. Shi, Supercond. Sci. Tech. 34, 035025 (2021).
TaC-2 Z. Cui, Y. Qian, W. Zhang, H. Weng, Z. Fang, Chinese Phys. Lett. 37, 087103 (2020).
Dirac-semimetal-NbC D. Yan, D. Geng, Q. Gao, Z. Cui, C. Yi, Y. Feng, C. Song, H. Luo, M. Yang, M. Arita, S. Kumar, E. F. Schwier, K. Shimada, L. Zhao, K. Wu, H. Weng, L. Chen, X. J. Zhou, Z. Wang, Y. Shi, and B. Feng Phys. Rev. B 102, 205117 (2020).
MoC A. Huang, A. D. Smith, M. Schwinn, Q. Lu, T. Chang, W. Xie, H. Jeng, and G. Bian, Phys. Rev. Materials 2, 054205 (2018).
Nodalsemimetal  N. Karn, M. Sharma, P. Sharma, G. Gurjar, S. Patnaik, and V. Awana, J. Supercond. Novel Magn. 34, 2717 (2021).
Fermisurface-nesting C. Li, N. K. Ravichandran, L. Lindsay, and D. Broido, Phys. Rev. Lett. 121, 175901 (2018).
ScS Arushi, R. K. Kushwaha, D. Singh, A. D. Hillier, M. S. Scheurer, and R. P. Singh, Phys. Rev. B 106, L020504 (2022).
Mo/WC R. H. Willens, E. Buehler, and B. T. Matthias, Phys. Rev. 159, 327 (1967).
3d-carbides N. J. Szymanski, I. Khatri, J. GAmar, D. Gall, S. V. Khare, J. Mater. Chem. C 7, 12619 (2019).












ReC4 H. Y. Gou, L. Hou, J. W. Zhang, and F. M. Gao, Appl Phys Lett 92, 241901 (2008).
ReC M. Kavitha, G. S. Priyanga, R. Rajeswarapalanichamy, K. Iyakuttti, Int. J. Refract. Met. Hard Mater. 52, 219 (2015).
Re4C Z. Zhao, L. Xu, L. M. Wang, B. Xu, M. Wang, Z. Liu, and J. He, Comput. Mater. Sci. 50, 1592 (2011).
ReC1 S. V. Popova and L. G. Boiko, High Temp. High Pressures 3, 237 (1971).
ReC2 Z. Zho, L. Cui, L. M. Wang, B. Xu, Z. Liu, D. Yu, J. He, X. F. Zhou, H. T. Wang, and Y. Tian, Cryst. Growth Des. 10, 5024 (2010).

ReC-WC P. Hu, X. Xie, L. Bai, R. Zhang, X. Zhang, J. Sun, H. Dong, M. Wen, and F. Wu, J. Appl. Phys. 131, 165102 (2022).
Mo/WReC2 A. C. Lawson, J. Less-Common. Met. 23, 103 (1971).
ReWC2-1 A. M. Tehrani, A. O. Oliynyk, M. Parry, Z. Rizvi, S. Couper, F. Lin, L. Miyagi, T. D. Sparks, and J. Brgoch, J. Am. Chem. Soc. 140, 9844 (2018).
ReWC2-2 S. G. Baird, M. Liu, H. M. Sayeed, and T. D. Sparks, arXiv:2202.02380, (2022).
FULLPROF J. Rodríguez-Carvajal, Physica B 192, 55 (1993).



ReC-latticeparameters S. V. Popova. Acta Crystallogr. Sect. A Cryst. Physics, Diffraction, Theor. Gen. Crystallogr. 31, 99 (1975).
WC1-x Z. Abdullaeva, E. Omurzak, C. Iwamoto, H. Okudera, M. Koinuma, S. Takebe, S. Sulaimankulova, and T. Mashimo, RSC Adv. 3, 513 (2012).


demagnetisation-factor A. Aharoni, J. Appl. Phys. 83, 3432 (1998).
BG-Model G. Grimvall, The Electron-Phonon Interaction in Metals (North-Holland, Amsterdam, 1981).
n A. Bid, A. Bora, and A. K. Raychaudhuri, Phys. Rev. B 74, 035426 (2006).
Tin1 M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996).
Tin2 T. Klimczuk, F. Ronning, V. Sidorov, R. J. Cava, and J. D. Thompson, Phys. Rev. Lett. 99, 257004 (2007).
Tin3 D. Singh, J. A. T. Barker, A. Thamizhavel, D. McK. Paul, A. D. Hillier, and R. P. Singh, Phys. Rev. B 96, 180501(R) (2017).
WHHM1 N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. 147, 295 (1966).
WHHM2 E. Helfand and N. R. Werthamer, Phys. Rev. 147, 288 (1966).
Pauli1 B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962).
Pauli2 A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).
Gi-1 G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994).
Gi-2 A. E. Koshelev, K. Willa, R. Willa, M. P. Smylie, J. K. Bao, D. Y. Chung, M. G. Kanatzidis, W. K. Kwok, and U. Welp, Phys. Rev. B 100, 094518 (2019).
Gi-3 O. Prakash, A. Thamizhavel, and S. Ramakrishnan, Supercond. Sci. Technol. 28, 115012 (2015).
el-ph W. L. McMillan, Phys. Rev. 167, 331 (1968).
BCS H. Padamsee, J. E. Neighbor, and C. A. Shiffman, J. Low Temp. Phys. 12, 387 (1973).
5-equations D. A. Mayoh, J. A. T. Barker, R. P. Singh, G. Balakrishnan, D. McK. Paul, and M. R. Lees, Phys. Rev. B 96, 064521 (2017).
CS/US Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, W. N. Hardy, R. Kadono, J. R. Kempton, R. F. Kiefl, S. R. Kreitzman, P. Mulhern, T. M. Riseman, D. L. Williams, B. X. Yang, S. Uchida, H. Takagi, J. Gopalakrishnan, A. W. Sleight, M. A. Subramanian, C. L. Chien, M. Z. Cieplak, G. Xiao, V. Y. Lee, B. W. Statt, C. E. Stronach, W. J. Kossler, and X. H. Yu, Phys. Rev. Lett. 62, 2317 (1989).
Tf A. D. Hillier and R. Cywinski, Appl. Magn. Reson. 13, 95 (1997).
Mo-WReB Y. Cui, J. Wu, B. Liu, Q. Zhu, G. Xiao, S. Wu, G. Cao, and Z. Ren, J. Alloy. Compd. 832, 154855 (2020).




