The XRD patterns for the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs are displayed in Figure 1A. For all x values, the major diffraction peaks can be well indexed on a cubic lattice with the P m 3 ¯ n space group, indicative of a dominant A15 phase. With increasing x, the (004) peak shifts toward higher 2θ values. This points to a decrease of the a-axis with the increase of V content, in consistent with its smaller atomic radius compared with those of Nb and Mo [22]. In addition to the A15 phase, small impurity peaks are observed in the vicinity of main (102) diffraction and probably comes from the NbAl₂-type sigma phase [18]. In the A15 structure, there are two crystallographic sites (0, 0, 0) and (0.25, 0, 0.5). Following Reference [18], all the five constituent elements are assumed to be distributed randomly on these sites for the structural refinement (see the inset of Figure 1A), and their occupancies are fixed by the stoichiometry. This assumption is based on the previous studies of binary A15 compounds, which show that the antisite disorder is the most common point defects [23]. In Nb₃Sn, it has been argued that the Nb and Sn atoms occupy randomly the two sites after a certain period of mechanical milling [24]. The refinement profiles are shown in Figures 1B–D and the statistics are listed in Table 1. Both the difference plot and R wp ( R p ) factor indicate a reasonably good agreement between the observed and calculated XRD patterns, which supports the validity of the employed structural model. Note that a more definitive conclusion requires atomic-level spectroscopies in future. The refined lattice parameter a = 5.0324, 5.0130, and 4.9848 Å for x = 0, 5 and 10, respectively, close to those of the A15-type V-Nb-Mo-Al-Ga HEAs. Figures 2A–C show the typical SEM images for the HEAs, all of which appear to be dense and homogeneous. Indeed, EDX elemental mapping reveals the uniform distribution of V, Nb, Mo, Ir, and Pt, and, as an example, the results for x = 0 are shown in Figures 2D–H. Furthermore, the EDX measurements allow us to determine the chemical compositions to be V_7.1Nb_33.8Mo_37.1Ir_10.7Pt_11.3, V_17.0Nb_28.9Mo_29.2Ir_11.2Pt_13.7 and V_26.3Nb_25.4Mo_26.0Ir_10.5Pt_11.8 for the HEAs with x = 0, 5, and 10, respectively. These agree well with the nominal compositions within the experimental error of ±2.5 at%.

Figures 3A,B show the temperature dependencies of resistivity (ρ) and magnetic susceptibility (χ) for the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs, respectively. For each x value, a sharp drop in ρ and strong diamagnetic χ are observed, signifying a superconducting transition. As indicated by the vertical dashed line, the midpoint of ρ drop coincides well with the onset of diamagnetic transition. By this criterion, T _c is determined to be 5.18, 4.49, and 3.61 K for the HEAs with x = 0, 5, and 10, respectively. Below T _c, there is a clear bifurcation between the zero-field cooling (ZFC) and field cooling (FC) χ data measured under an applied field of 1 mT, which is characteristic of a type-II superconductor. At 1.8 K, the χ ZFC data correspond to superconducting shielding fractions ranging from 101 to 174%. Although the demagnetization effect is difficult to correct due to irregular sample shapes, these large values suggest bulk superconductivity in these HEAs.

To confirm the bulk nature of superconductivity, the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs were further characterized by specific heat ( C p ) measurements, whose results are shown in Figure 4. As can be seen in Figures 4A, a C _p jump is indeed detected around T _c for these HEAs. Above T _c , the data are analyzed by the Debye model C p / T = γ + β T 2 + δ T 4 , (1) where γ and β(δ) are the Sommerfeld and phonon specific heat coefficients, respectively. With β, the Debye temperature Θ D is calculated as Θ D = ( 12 π 4 R / 5 β ) 1 / 3 , (2) where R is the molar gas constant 8.314 J/molK². This gives γ = 4.59, 4.94, and 5.03 mJ/molatomK², and Θ D = 419, 440, and 393 K for x = 0, 5, and 10, respectively. Figure 4B shows the normalized electronic specific heat C _el/γT after subtraction of the phonon contribution. For all HEAs, the ΔC _el/γT are significantly smaller than the BCS value of 1.43 [25]. Nevertheless, the C _el/γT data can still be fitted by a modified BCS model or the α-model [26] with α = 1.39, 1.41 and 1.56 for x = 0, 5 and 10, respectively, where α = Δ₀/T _c and Δ₀ is the gap size at 0 K. These results suggest that the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs are BCS-like superconductors in the weak coupling regime. This is corroborated by their electron-phonon coupling constants λ ep in the range of 0.55–0.59, as calculated using the inverted McMillan formula [27], λ e p = 1.04 + μ ∗ ⁡ ln ( Θ D / 1.45 T c ) ( 1 − 0.62 μ ∗ ) ln ( Θ D / 1.45 T c ) − 1.04 , (3) with μ * = 0.13 being the Coulomb repulsion pseudopotential. In passing, it is pointed out that the decrease in T _c with increasing x is accompanied by the decrease in λ ep but the increase in γ. Hence the T _c in the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs is mainly governed by the electron-phonon coupling strength rather than the density of states at the Fermi level. In passing, it is worth noting that the T _c, γ and λ ep values of the V–Nb–Mo–Pt–Ir HEAs are very similar to those of the ( V 0.5 Nb 0.5 )_3−x Mo _x Al 0.5 Ga 0.5 HEAs for x ≥ 1.2 [18], pointing to a common phonon-mediated pairing mechanism.

The upper critical fields B c 2 of these HEAs were investigated by resistivity measurements under magnetic fields. As an example, the result for the HEA with x = 0 is shown in Figure 5A. The resistive transition is gradually suppressed to low temperatures as the field increases. For each field, the T _c is determined using the same criterion as above, and the obtained B c 2 vs. T phase diagrams are displayed in Figure 5B. Extrapolating the B c 2 (T) data to 0 K using the Wathamer-Helfand-Hohenberg model [28] yields B c 2 (0) = 6.4, 5.7 and 4.4 T for the HEAs with x = 0, 5, and 10, respectively. These values are well below the corresponding Pauli limiting fields [29] of ∼9.6, ∼8.4, and ∼6.7 T, suggesting that B c 2 in these HEAs is orbitally limited. In addition, the Ginzburg–Landau coherence lengths ξ GL can be calculated using the equation ξ GL ( 0 ) = Φ 0 / 2 π B c 2 ( 0 ) , (4) where Φ 0 = 2.07 × 10 − 15  Wb is the flux quantum. This yields ξ GL = 7.2, 7.6 and 8.7 nm for the HEAs with x = 0, 5, and 10, respectively. The above results are summarized in Table 1.

Figure 6 shows the VEC dependence of T _c for the V_5+2x Nb_35−x Mo_35−x Ir₁₀Pt₁₅ HEAs, together with the data for A15-type V-Nb-Mo-Al-Ga HEA [18] and binary [20] superconductors for comparison. One can see that superconductivity in all these materials occurs near the VEC values of 4.7 and 6.5, consistent with the expectation from the Matthias rule [30]. Compared with the V–Nb–Mo–Al–Ga HEAs, the V–Nb–Mo–Ir–Pt HEAs have higher VEC values in the range of 6.4–6.5 and their VEC dependence of T _c is in the opposite trend, increasing monotonically with the increase of VEC. Nevertheless, the maximum T _c is considerably lower for the V–Nb–Mo–Ir–Pt HEAs than for the V-Nb-Mo-Al-Ga ones. This indicates that optimal VEC for T _c in A15-type HEA superconductors is around 4.7, which is reminiscent of the case in binary A15 compounds [20]. Moreover, for similar VEC values, the T _c values for V–Nb–Mo–Al–Ga and V–Nb–Mo–Ir–Pt HEAs are always no more than half those of the binary compounds. It is thus reasonable to speculate that the upper limit of T _c for A15-type HEA superconductors is about one-half the highest T _c in binary A15 superconductors.