In the absence of an applied magnetic field, the electrical resistivity of a superconductor tends to zero upon cooling to a critical temperature, T_c [ 1 ]. As the temperature falls to T_c, and when the applied magnetic field is weaker than a critical field, H_c, the magnetic flux is completely expelled from the superconductor, and a perfect diamagnetic state is formed [ 2 ]. Superconductors with these properties have been widely studied and applied in various applications.

Among elements in nature, Nb has the highest T_c (9.25 K). Conventional binary Nb-based alloys, such as Nb-Ti (~10 K), Nb-Zr (~11 K), Nb₃Sn (18 K), and Nb₃Ge (23 K), have even a higher T_c than Nb [ 3 ]. These T_c values are too low to enable the elements to be used under ambient conditions. Therefore, many attempts have been made to raise T_c. For example, researchers have used various heat treatments to control the formation of the superconducting phase of, e.g., Nb-Zr, [ 4 ]. In Nb-Ti [ 5 ], as another example, third elements, Al & Mo, have been used to generate 4.6 to 4.8 valence electrons per atom, e/a [ 6 ]. According to Matthias’ empirical rule, the highest T_c is obtained at the optimal e/a ratio for which the composition chooses, e.g., Mo, as a third element to reduce T_c in Nb-Ti [ 5 ]. However, adding Al promotes the formation of a phase of Nb-rich and increases T_c [ 5 ]. However, the T_c in ternary systems barely exceeds that of binary Nb-Ti. Accordingly, the effectiveness of such approaches to increase T_c is limited. Moreover, Matthias’ empirical rule may not be applicable to multi-phase alloys. Difficulties still exist.

High-entropy alloys (HEAs) were proposed in 1995. These alloys were obtained by vacuum arc remelting [ 7 , 8 ]. The efforts to design and prepare HEAs have opened up yet another frontier in alloy research. They have been defined as alloys that contain principal elements of between five and 13 (inclusive), and have atomic concentrations of between five and 35 at% (inclusive). Their main characteristics are high solubility, a large degree of lattice distortion, a large point lattice defect effect, slow atom diffusion, and a composite property effect (the so-called cocktail effect). Thermodynamically, the high-entropy effect, is a TS factor and a physical effect overcomes the enthalpy effect, which is an H factor and a chemical effect, increasing the solubility of each of the principal elements, promoting the generation of a random simple pseudo-unitary solid solution, such as with the A1, A2, or A3 crystal structure. This tendency to form a simple random solid solution thermodynamically enables the slow diffusion of atoms in the working processes, as occurs in thermal treatments. Hence, a nanostructure can be easily obtained. The high solubility of constituent elements makes the random solid solution highly ductile and hard. The HEA can simultaneously have the properties that conventional alloys with fewer multi-principal-element alloys cannot easily have simultaneously, such as heat resistance, anticorrosion, anti-oxidation, and precipitation hardening at high temperatures. Therefore, HEAs have great potential in the development of a range of applications [ 9 ]. HEAs have superior physical properties such as electrical conductivity, magnetic permeability, thermal conductivity, and hydrogen storage [ 10 , 11 ].

This study will explore superconductivity in HEAs. Multi-principal elements are added to modify their microstructure to study the effect of adding elements on the superconducting behaviors―zero electrical resistivity and diamagnetism―of simple pseudo-unitary BCC alloys [ 11 ]. The selected materials were equimolar NbTaTiZr-based HEAs. The results are compared with those for conventional superconductors.

Superconductivity in conventional alloys has been microscopically explained by the BCS, which was proposed by Bardeen, Cooper, and Schrieffer in 1957 [ 12 ]. The central point of BCS is that electron Cooper pairs that are formed by electron-electron interaction under the influence of lattice phonon as the temperature falls below T_c generate no electrical resistivity. The critical temperature for superconductivity is given by, [ 13 ]

T c = 1.14 θ D exp ( − 1 λ ) , (1)

λ = D ( ε F ) V , (2)

where λ is a superconducting coupling constant; D(ε_F) is the electron density of states at the Fermi level; θ D (=ħω/k_B) is the Debye temperature; k_B is Boltzmann’s constant; ω is the frequency of lattice vibration, and V is of the interaction between electrons in Cooper pairs. BCS applies only in weak coupling condition, i.e., 0 < λ ≪ 1 . In 1968, McMillan developed another expression for T_c, which considers the electron-phonon interaction, and is affected by both weak and strong couplings. This T_c is given by [ 14 ]

T c = θ D 1.45 exp [ − 1.04 ( 1 + λ ) λ − μ * ( 1 + 0.62 λ ) ] , (3)

where θ_D is Debye temperature; λ is a coupling constant for the interaction between electrons and phonons, and μ^* is the repulsive coupling interaction constant in the interaction between electrons. McMillan also found that, for most metals, μ^* ≈ 0.13. Accordingly, Equation (3) can be simplified as:

T c ≈ θ D 1.45 exp [ − ( 1 + λ ) λ ] , (4)

λ = D ( ε F ) 〈 I 〉 2 M 〈 ω 〉 2 , (5)

where I is the electron-phonon matrix element; ω is the phonon frequency in the metal lattice, and M is the mass of a metal ion. From Equation (5), a larger λ, associated with higher D(ε_F), a smaller ω or a smaller θ_D, are associated with a higher T_c. For this reason, alloying elements are added. One paper on superconductivity in Nb-Cu noted that, as the temperature falls to T_i, the silky network of the superconducting phase begins to allow the Cooper pairs to pass the alloy, generating superconductivity [ 15 ]. As the temperature falls further to T_f, if the coherence length is large enough, then the Cooper pairs generate an integrated superconducting path by the proximity effect. Therefore, determining whether superconducting HEAs exhibit the proximity effect and whether they are type II superconductors is essential. Some appropriate theories are used to calculate the coherence length (ξ) and the Ginzburg-Landau parameter (κ) from the carrier density [ 16 ]:

n = I B V H q t , (6)

where n is the carrier density at 5 K; I is the current; B is the applied magnetic field; V_H is the Hall voltage; t is the thickness of sample; and q is the electric charge. After resistance vs. temperature (R-T) measurements were made to determine the electrical resistivity, the Drude model was used to calculate the electron relaxation time τ [ 16 ].

ρ = m n q 2 τ , (7)

where ρ is the 300 K-resistivity; m is the electron mass; n is the carrier density; and q is the electric charge. The Fermi velocity u_F can also be obtained from the carrier density using Equation (8) [ 16 ].

υ F = ( ℏ m ) ( 3 π 2 n ) 1 3 . (8)

The Fermi velocity and relaxation time can be used to determine the mean free path of the electrons using Equation (9) [ 16 ].

l = υ F τ . (9)

Equations ((10) & (11)) yield the superconducting energy gap at 0 K, Δ(0), and the intrinsic coherence length, ξ₀, respectively [ 17 ].

Δ ( 0 ) = 1.76 k B T c . (10)

ξ 0 = ℏ υ F π Δ ( 0 ) , (11)

where k_B is Boltzmann’s constant; T_c is the superconducting critical temperature; and ħ is Planck’s constant over 2π.

In the mixed state of type II superconductors, each Abrikosov vortex has a core radius, ξ, at normal conducting (NC) state, surrounded by an outer radius, λ, of superconducting current. Each vortex carries a quantized flux, Φ₀ = 2.0678 × 10⁻⁷ G cm². As the applied magnetic field increases to the lower critical field, H_c1, vortices arise. Hence, the flux Φ₀ can be expressed as Equation (12) [ 16 ]:

ϕ 0 ≅ π λ 2 H c 1 . (12)

The Ginzburg-Landau parameter, κ, can thus be obtained and used to identify the type of a superconductor by applying Equation (13) [ 3 ].

κ = λ ξ . (13)

The existence of superconductivity in HEAs was demonstrated and the coherence length is shown to be of the order of several hundred Ǻ, equal to that of binary Nb-Zr and Nb-Ti. Therefore, there is also a proximity effect like that in Nb-Zr and Nb-Ti, is suggested to exist in superconducting HEAs. The calculated κ value is 14.32. This value, taken with results of magnetic measurements, reveals that superconducting HEAs are type II superconductors. Notably, Fe-containing NbTaTiZrFe has T_c = 6.87 ± 0.03 K. The highest obtained T_c = 8.46 ± 0.02 K of NbTaTiZrGe [ 18 ] is close to that of element Nb of 9.25 K, the two values are near. This result is not good enough, but, owing to the many degrees of freedom in the design of alloys, superconducting HEAs may open up a new avenue for research and applications.

JEOL 5410 SEM-energy-dispersed X-ray spectroscope (SEM-EDS) was used to observe the microstructure and analyze the composition of alloys. Shimadzu XRD 6000 diffractometer was used in X-ray diffraction with Cu-target (wavelength of 1.5405 Å), and operated at 30 kV and 20 mA. The scanning speed was 2˚/min with a 2θ range of 20 to 100 degrees.

To measure the transition temperature, specimens were cut in rectangles with a length of 8 mm and width of 3 mm, and ground using #400 SiC grinding paper to a thickness of 0.2 mm. The four-point method was used to plot the resistivity vs. temperature, R(T), curves at 4.2 K to 300 K. The accuracy of the temperature measurement was better than ±0.06 K. The critical transition temperature, T_c, is defined as the temperature at which the resistivity drops to 50% of the total resistivity. The interval of transition is defined by the formula ΔT_c = T_c(90%) − T_c(10%), where 90% and 10% indicate a drop in the resistivity to 90% and 10% of the original value, respectively, in the transition.

A SQUID setup was used to obtain both magnetic susceptibility vs. temperature, χ(T), and magnetization vs. applied magnetic field, M(H). χ(T) at temperatures in the range 2 K to 300 K were obtained using the zero-field cooling (ZFC) method in applied magnetic fields of 100 Oe. Two fixed temperatures, 5 K and 300 K, were used and the applied field was cyclically varied from 0 to 1 to 0 to −1 to 0 T, and then back to 1 T. The lower magnetic field, H_c1, is the field that induces deviation from the diamagnetism (χ = −1) line. When the demagnetization factor is considered, H_c1 is given by Equation (14):

H c 1 = H 1 − D , (14)

where D is demagnetization factor. The upper critical field, H_c2, is the smallest field that yields zero magnetization.

Hall measurements were made using a PPMS system (Quantum Design PPMS-9T) that was operated at 47 Hz, 50 mA AC, and from 9 T to −9 T at 10 K. Pt wires were attached using low-temperature Ag paste to the specimens to measure resistivity and to make the Hall measurements. The attachments were baked at 50˚C to ensure an Ohmic contact. Carrier density was determined from the Hall voltage data.

Table 1 lists selected alloys, their phases, and their compositions. Both as-cast (C state) and homogenized (H state) specimens were used. Some designations are given. They are equal-molar C4 & H4 for NbTaTiZr, C5 & H5 for NbTaTiZrHf, C6 & H6 for NbTaTiZrHfV, and C7 & H7 for NbTaTiZrHfVMo. All specimens were melted and cast in a remelter in an atmosphere of Ar using a water-cooled Cu hearth. As-cast specimens were melted once, and then turned and melted three more times to ensure their homogeneity. Samples were homogenized in a quartz tube that was filled with argon gas at 1100˚C for 1 d and then quenched in water.

Figure 1 shows back-scattered electron images (BEI). Table 1 lists energy-dispersed X-ray spectroscopic (EDS) data for both as-cast and homogenized specimens. White Nb-Ta-rich dendrites and black Ti-Zr-rich interdendrites are observed in C4, and white Nb-Ta-rich dendrites and black Zr-Hf-rich interdendrites are observed in both C5 and C6. XRD patterns (Figure 2) reveal that alloys C4 to C6 are simple

pseudo-unitary A2 random BCC solid solutions with lattice parameters of 3.3728 Å, 3.4051 Å, and 3.3567 Å, respectively. H4 to H6 have a matrix, grown from the dendrites, and they contain less interdendrite than the C4 to C6, respectively. The crystal structures and lattice parameters of the as-homogenized samples are almost the same as those of the as-cast samples. Alloys with added Mo, Fe, Ge, Hf, Si, V, and/or W are used to examine the dependence of T_c on the e/a ratio and to verify Matthias’ empirical rule. These are NbTaMoW, NbTaMoVW, NbTaTiZrHfVMo (C7 BCC1 & H7 BCC with lattice parameters of 3.2840 & 3.2711 Å, respectively), NbTaTiZrFe, NbTaTiZrGe, NbTaTiZrGeV, NbTaTiZrSiV, NbTaTiZrSiGe, and NbTaTiZrSiGeV.

Superconductivity was observed in NbTaTiZr-based HEAs with added Fe, Ge, Hf, Si, and/or V. The SC behavior of HEAs in this study depends on both the composition distribution of Nb + Ta and the degree of lattice distortion that is caused by the change in the number of contained elements, n. As n increases from four to seven, the Nb + Ta content in the SC white phases falls, reducing T_c. In multiphase HEAs, the e/a ratio alone cannot be used to evaluate accurately T_c, as in the manner of the Matthias rule. This study uses analysis of R(T), and SQUID and Hall measurements to demonstrate that the HEAs in this study are type II SCs.

We acknowledge the financial support of the National Science Council, Taiwan, under No. 100-2221-E-007-110-MY3.