Superconductivity in a New High-Entropy Alloy (NbTi)$_{0.67}$(MoHfV)$_{0.33}$

Abstract

The structural and superconducting properties of a new niobium- and titanium-rich high-entropy alloy (NbTi)$_{\text {0.67}}$(MoHfV)$_{\text {0.33}}$ were determined. The alloy was synthesized by arc melting and its physical properties were characterized by means of X-ray powder diffraction, energy dispersive X-ray spectroscopy, magnetization, electrical resistivity and specific heat measurements. Experimental data revealed that the (NbTi)$_{\text {0.67}}$(MoHfV)$_{\text {0.33}}$ crystalizes in body-centered cubic structure and exhibits type-II superconductivity below about 5 K. In addition, electronic structure calculations were performed using the Density Functional Theory (DFT) method. Their results suggest that in case of HEAs, the influence of the local atomic configuration on their electronic structure in the energy region close to E$_{\mathrm {\text {F}}}$ is rather small or even negligible.

1 Introduction

High-entropy alloys (HEAs) are a new class of materials with unique physical and chemical properties. HEA can be defined as a solid solution (an alloy) with a simple crystal structure consisting of five or more principal elements along with individual contribution between 5 and 35 at. pct.^[1,2] Since it has been reported that many HEAs possess excellent mechanical, magnetic and corrosion resistant properties, they are being considered for a wide range of functional applications.^[3,4]

The first HEA superconductor has been discovered in 2014.^[5] It is the high-entropy alloy with the composition Ta$_{34}$Nb$_{33}$Hf$_{8}$Zr$_{14}$Ti$_{11}$ (in at. pct) crystallizing in body-centered cubic structure (bcc), being a conventional type-II superconductor with the transition temperature $T_{\rm{c}} \approx$ 7.3 K, the lower critical field $\mu _{\rm{0}} H_{\rm{c1}} \approx$ 32 mT, and the upper critical field $\mu _{\rm{0}} H_{\rm{c2}} \approx$ 8.2 T. Since that discovery, researchers have identified many new superconducting HEAs which are mainly composed of 3d, 4d and 5d transition metals.^{[6,7,8,9,10,11]} Unfortunately, most of these superconductors have lower $T_{\rm{c}}$ and $H_{\rm{c2}}$ than commercially used Nb–Ti and Nb$_3$Sn materials.^[12] Despite the relatively low critical parameters compared to currently intensively studied hydrogen-rich superconductors with $T_{\rm{c}}>$ 200 K under high pressures,^[13,14,15] the further study of the HEA superconductors seems to be attractive due to their application potential, since they are easy to synthesize at low cost, and capable of superconducting at atmospheric pressure.

Recently, our group reported on the discovery of niobium- and titanium-rich high-entropy alloy Nb$_{34}$Ti$_{33}$Zr$_{14}$Ta$_{11}$Hf$_{8}$.^[16] The alloy crystallizes in bcc structure and exhibits type-II superconductivity with $T_{\rm{c}} \approx$ 7.5 K and $\mu _{\rm{0}} H_{\rm{c2}} \approx$ 12.2 T. This finding indicates that Nb$_{34}$Ti$_{33}$Zr$_{14}$Ta$_{11}$Hf$_{8}$ has one of the highest upper critical field in the family of HEA superconductors. The relatively high value of $H_{\rm{c2}}$ parameter observed in this HEA is probably a consequence of the fact that the alloy contains 33 at. pct of titanium, while in previously studied superconducting HEAs, the titanium concentration is typically less than 20 at. pct.

Taking the above into account, we decided to synthesise a new niobium- and titanium-rich high-entropy alloy (NbTi)$_{0.67}$(MoHfV)$_{0.33}$. Structural and physical properties of the prepared material were determined using various experimental techniques. Additionally, electronic structure calculations were performed using two complementary approaches within density functional theory. The obtained results are discussed and compared with data available in the literature for other HEA superconductors.^{[6,7,8,9,10,11,16]}

2 Materials and Methods

The (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ sample was made of high-purity metals (at least 99.9 pct purity) which were in the form of slugs, chips or wires. The synthesis was conducted using an arc furnace in an argon atmosphere. The 1 g ingot of material was remelted four times to improve its homogeneity. In the furnace chamber, titanium was placed as a getter, which was heated each time before the material was remelted to purge the argon atmosphere of residual oxygen and nitrogen. After the complete procedure, the total weight loss of the obtained material did not exceed 0.2 pct.

The crystal structure of the prepared sample was studied at room temperature by X-ray powder diffraction (XRD) using a PANalytical X’pert Pro diffractometer with CuK$_\alpha$ radiation. The obtained XRD pattern was analyzed by the Rietveld method implemented in the FullProf software package (version January-2023).^[17] The homogeneity and chemical composition of the alloy were verified by energy dispersive X-ray spectroscopy (EDXS) using a FESEM FEI Nova NanoSEM 230 scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer on a polished surface of the cut specimen.

Physical properties including AC and DC magnetization, specific heat as well as electrical resistivity as a function of temperature and applied magnetic field were investigated using a Quantum Design Physical Properties Measurements System (PPMS) platform. The AC magnetic susceptibility was measured with an applied AC field of 1 mT amplitude and 1 kHz frequency. The AC and DC magnetization measurements were performed in the presence of applied magnetic fields up to 9 T in the range from 2 K to 6 K. The shape of the sample for magnetic susceptibility measurements resembled a cuboid with dimensions of 1.4 $\times$ 1.0 $\times$ 5.0 mm$^3$. Using the formula presented in paper^[18]:

$$\begin{aligned} N = \frac{4ab}{4ab+3c(a+b)}, \end{aligned}$$

(1)

the value of the effective demagnetization factor was estimated as $N = 0.13(1)$. The resistivity was measured using a four-contact method by passing an alternating current through the sample with an amplitude of up to 20 mA and a frequency of 33 Hz. The measurement of $\rho$ was performed in the presence of applied magnetic fields up to 7 T in the range from 2 K to 300 K. The bar-shape sample used in resistivity measurements had dimensions of 1.5 $\times$ 1.0 $\times$ 1.0 mm$^3$. For the specific heat measurements, the sample was cut into cube with a side of 1.0 mm. The data were collected using the thermal relaxation technique and the two-tau model. The heat capacity of the sample platform with a very small amount of Apiezon N vacuum grease was determined and subtracted prior to the data analysis.

Electronic structure calculations were performed using two complementary approaches. The first one was based on the Korringa-Kohn-Rostoker Coherent Potential Approximation (KKR-CPA) formalism.^{[19,20,21,22,23]} In this study, the KKR-CPA method implemented in the AkaiKKR (machikaneyama) package^[24,25,26] was used. To construct the muffin-tin crystal potential in the semi-relativistic approach, the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE) was used.^[27,28,29] The angular momentum cutoff was set to $l_{\rm{max}}$ = 3 and 5216 k-points were used to sample the irreducible part of the Brillouin zone during the self-consistent cycle and density of states calculations. The Atomic Sphere Approximation (ASA) was applied in all calculations. The lattice parameter $a_{\mathrm{KKR-CPA}}$ = 3.34 Å, the bulk modulus $B_{\mathrm{KKR-CPA}}$ = 142 GPa and its derivative $B^{\prime}_{\mathrm{KKR-CPA}}$ = 4.2 of the crystal (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ were determined using the Birch-Murnaghan equation of state.^[30] Independently, the electronic structure of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ system was determined by supercell DFT calculations in the plane-wave function basis implemented in the Vienna Ab initio Simulation Package (VASP).^[31,32,33] The electron exchange-correlation interactions were treated at the Generalized Gradient Approximation (GGA) level using the PBE functional form. The electron ion-core interactions were represented by the Projector Augmented Wave (PAW) potentials.^[34,35] A plane-wave basis set with a kinetic energy cutoff of 400 eV was used. The supercell approach requires a well-defined position of all constituent atoms thus three different random atomic configurations were considered. In order to reproduce the chemical composition of the studied HEA, all constructed supercells contain 54 atoms with 18 Ti, 18 Nb, 6 Mo, 6 Hf and 6 Zr atoms randomly placed at the nodes of the body-centered cubic $3\times 3\times 3$ supercell. The lattice parameters, derived using the Birch-Murnaghan equation of state, for all three configurations close to $a_{\rm{PAW}}$ = 3.263 Å  $B_{\rm{PAW}}$ = 143 GPa and $B^{\prime}_{\rm{PAW}}$ = 4.03 are in good agreement with those obtained by the KKR-CPA method ($a_{\mathrm{KKR-CPA}}$ = 3.34 Å, $B_{\mathrm{KKR-CPA}}$ = 142 GPa).

3 Results and Discussion

3.1 Chemical Composition and Crystal Structure

Figure 1 shows the EDXS elemental mapping and SEM micrograph of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$. The images show an even distribution of elements over almost the entire surface of the sample. The determined atomic concentrations of Nb 31.7(7) pct, Ti 32.9(7) pct, Mo 12.8(6) pct, Hf 11.5(5) pct, V 11.2(5) pct are almost the same as the nominal composition: Nb 33.5 pct, Ti 33.5 pct, Mo 11.0 pct, Hf 11.0 pct, V 11.0 pct. However, small areas of the sample exhibit a slightly higher molybdenum content (see the bright stripes in bottom left image in Figure 1). The chemical composition of these areas is close to Nb 33.3(7) pct, Ti 30.6(7) pct, Mo 16.0(7) pct, Hf 10.3(4) pct, V 9.8(4) pct. This observation can be supported by the XRD pattern shown in Figure 2.

Fig. 1

EDXS elemental mapping and SEM micrograph (bottom right image) of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$

Fig. 2

Powder X-ray diffraction pattern for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ with a result of Rietveld refinement. Red circles are experimental data, black line is a theoretical curve, blue line is the difference between the two and green dashes show positions of the Bragg reflections described by the corresponding Miller indices (Color figure online)

The XRD pattern shown in Figure 2 revealed that the (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ alloy consists of two crystalline phases. Both of these can be described by a body-centered cubic (bcc) crystal structure (space group ${Im{\overline{3}}m}$, W-type structure). The values of the weighted profile R-factor $R_{wp}=13.4$ and the expected R-factor $R_{exp}=6.4$ and the difference between the experimental points and the theoretical curve prove the good quality of the Rietveld refinement. It showed the main phase is approximately 91 wt pct of the sample and its lattice parameter is $a = 3.267(1)\, \mathrm {\text{\AA }}$, which is in agreement with the composition-averaged value $a_{th} = 3.274 \,\mathrm {\text{\AA }}$ calculated using the Vegard’s law^[36] and with the values obtained using DFT calculations, in particular, the one derived using the Birch-Murnaghan equation of state ($a_{\mathrm{KKR-CPA}}$ = 3.34 Å, $a_{\rm{PAW}}$ = 3.263 Å). The secondary phase, which accounts for about 9 wt pct of the sample, exhibits a lattice parameter of $3.276 \,\mathrm {\text{\AA }}$. The value of the lattice parameter of the main phase is slightly smaller than most of the reported transition metal-based HEAs crystallizing with a bcc structure.^{[5,16,37,38,39,40]} The occurrence of a secondary phase in high entropy alloys is a common issue.^[10,41]

3.2 Electrical Resistivity

Figure 3 shows results of the measurement of electrical resistivity $\rho$ of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ alloy as a function of temperature. As the temperature decreases, the resistance decreases from 146 $\mathrm {\mu \Omega \, cm}$ at room temperature to 132 $\mathrm {\mu \Omega \, cm}$ at temperature just above the transition to the superconducting state. The determined residual resistivity ratio RRR is close to 1.11, which is lower value than corresponding 1.2 for Nb$_{34}$Ti$_{33}$Zr$_{14}$Ta$_{11}$Hf$_{8}$,^[16] 1.4 for ${(\rm{NbTa})}_{0.67}{(\rm{MoHfW})}_{0.33}$^[8] and 2.54 for ${(\rm{NbTa})}_{0.67}{(\rm{MoWTh})}_{0.33}$.^[11] This finding reflects the extremely high degree of chemical and structural disorder in the studied HEA.

Fig. 3

Electrical resistivity as a function of temperature performed for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ in zero magnetic field. The inset shows the low-temperature resistivity in nominal applied magnetic fields $\mu _{\rm{0}}H$. Solid lines were added to guide the eye

The critical temperature was defined as the temperature of the first zero point below the sharp drop in resistance $\rho ^{\mathrm {0\,pct}}$. In the case of measurement in zero applied magnetic field, the critical temperature is $T_{\rm{c}}^{\rho ^{\mathrm {0\,pct}}} \approx 5.0$ K. As the intensity of the applied magnetic field increases, the critical temperature of the superconducting transition systematically decreases. At $\mu _{\rm{0}}H = 7$ T, no transition is observed in the temperature range studied.

Worth mentioning is valence electron concentration (VEC),^[10] which is defined as:

$$\begin{aligned} \rm{VEC} = \sum _i c_i \rm{VEC}_i, \end{aligned}$$

(2)

where $\rm{VEC}_i$ is the VEC value of the i-th element and $c_i$ is the atomic concentration of the i-th element. Using Eq. [2] and the EDXS data, it can be calculated that the VEC of the main phase is 4.68, while that of the secondary phase it is 4.75. According to the papers^[6,10,39] the maximum of $T_{\rm{c}}$ is expected at VEC $\approx$ 4.7. This suggests that the transition to the superconducting state seen in Figure 3 is related to the main phase, since the secondary phase should have a lower critical temperature.

3.3 Magnetic Properties

Figure 4 shows temperature dependence of AC susceptibility of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ measured in several applied external magnetic fields. Both the real $\chi ^{\prime}$ and imaginary $\chi ^{\prime\prime}$ parts of the susceptibility confirm the presence of bulk superconductivity in the alloy. $T_{\rm{c}}$ in the smallest external field is about 5 K and decreases with increasing magnetic field up to a value of 7 T, at which the transition is not longer observed in the temperature range available in the experiment.

Fig. 4

Real $\chi ^{\prime}$ and imaginary $\chi ^{\prime\prime}$ part of susceptibility of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ measured as a function of temperature in nominal applied magnetic fields $\mu _{\rm{0}}H$. Solid lines were added to guide the eye

The results of DC mass magnetization measurements carried out for the (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ alloy as a function of temperature are shown in Figure 5. The measurements were conducted in the zero-field-cooling (ZFC) regime in several external magnetic fields applied. Based on the mass magnetization value measured at the lowest temperature of 2.1 K in a nominal field of 10 mT, i.e. $-1.06 \, \mathrm {A \, m^2 \, kg^{-1}}$, the superconducting volume fraction (SVF) was estimated to be about 85(1) pct.

Fig. 5

Temperature dependence of mass magnetization $\sigma$ of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ measured in zero-field-cooling (ZFC) for various applied magnetic fields $\mu _{\rm{0}}H$. Solid lines were added to guide the eye

The field dependence of the mass magnetization of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ measured at several temperatures, shown in Figure 6, reveals that the alloy is a type-II superconductor with a small lower critical field $\mu _{\rm{0}}H_{\rm{c1}}$ (< 20 mT at 2 K). Comparable values of $\mu _{\rm{0}}H_{\rm{c1}}$ were reported for similar systems, i.e. 32 mT for Ta$_{34}$Nb$_{33}$Hf$_{8}$Zr$_{14}$Ti$_{11}$,^[5] 7 mT for ${(\rm{NbTa})}_{0.67}{(\rm{MoHfW})}_{0.33}$^[8] and $<15$ mT for Nb$_{34}$Ti$_{33}$Zr$_{14}$Ta$_{11}$Hf$_{8}$.^[16] At higher magnetic fields, the mass magnetization curves show a gradual collapse of the vortex state—a behavior characteristic of type-II superconductors. However, the determination of the upper critical fields $\mu _{\rm{0}}H_{\rm{c2}}$ is difficult due to the strong paramagnetic contribution to the total magnetization of the sample.

Fig. 6

Magnetic field dependence of DC mass magnetisation $\sigma$ at various temperatures measured for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$. Solid lines were added to guide the eye

3.4 Specific Heat

The temperature dependence of the specific heat (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ is shown in Figure 7. As can be seen, at temperatures above the SC transition, $C_{\rm{p}}(T)$ exhibits a behavior characteristic of metallic materials. The experimental curve can be described by the Debye model for lattice vibration with correction for anharmonicity^[42]:

$$\begin{aligned} C_{\rm{p}}(T) = \gamma T + \frac{9Rn}{1+\alpha T} \big (\frac{T}{\Theta _{\rm{D}}^{\rm{HT}}}\big )^3 \int _0^{\Theta _{\rm{D}}^{\rm{HT}}/T} \frac{x^4 e^x}{(e^x -1)^2} dx, \end{aligned}$$

(3)

where $\Theta _{\rm{D}}^{\rm{HT}}$ refers to the Debye temperature estimated from high temperature data, R is the universal gas constant, $n = 1$ is the number of atoms per formula unit (f.u.), and $\alpha$ stands for the parameter describing the anharmonicity of the system. The value of the Sommerfeld coefficient $\gamma$ can be obtained from fitting of the formula:

$$\begin{aligned} \frac{C_{\rm{p}}}{T} = \gamma + \beta T^2, \end{aligned}$$

(4)

to the experimental data just above $T_{\rm{c}} = 4.8(2)$ K (see the lower inset in Figure 7), which yields the lattice specific heat coefficient $\beta = 0.621(32) \,\mathrm {mJ \, K^{-4} \, mol^{-1}}$ and $\gamma = 7.41(10) \,\mathrm {mJ \, K^{-2} \, mol^{-1}}$. Using the $\beta$ parameter extracted from the this fit, the low-temperature Debye temperature $\Theta _{\rm{D}}^{\rm{LT}} = 315(5) \,\rm{K}$ was calculated via the formula:

$$\begin{aligned} \Theta ^{\rm{LT}}_{\rm{D}} = \big (\frac{12nR\pi ^4}{5\beta } \big )^{\frac{1}{3}}. \end{aligned}$$

(5)

As a result, the fit of Eq. [3] to the experimental data (see the solid line in the main panel in Figure 3) gave the parameters $\Theta _{\rm{D}}^{\rm{HT}} = 289(1) \,\rm{K}$ and $\alpha = 2.0(2) \times 10^{-4} \,\mathrm {K^{-1}}$.

Fig. 7

Temperature dependence of specific heat $C_{\rm{p}}$ for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$. The lower inset shows the dependence of $C_{\rm{p}}/T$ vs $T^{2}$, and the upper inset shows the dependence of $\ln ({C_{\rm{es}}/\gamma T_{\rm{c}}})$ vs $T^{-1}$ (see the text for details). Solid lines in the main panel and in the lower and the upper insets show fits of Eqs. [3], [4], and [8] to the experimental data, respectively

At the same time, the determined $\gamma$ parameter can be used to calculate the density of states (DOS) of conducting electron at the Fermi level $N(E_{\rm{F}}) = 3.14(4) \, \mathrm {st. \, eV^{-1} \, f.u.^{-1}}$, based on the formula:

$$\begin{aligned} \gamma = \frac{1}{3} \pi ^2 k_B^2 N_{\rm{A}} N(E_{\rm{F}}), \end{aligned}$$

(6)

where $k_B$ is the Boltzmann constant and $N_A$ is the Avogadro number.

By determining $T_{\rm{c}}$ and $\Theta ^{\rm{LT}}_{\rm{D}}$, the electron-phonon coupling constant $\lambda _{\mathrm {el-ph}}$ can be calculated using the McMillan relation:

$$\begin{aligned} \lambda _{\mathrm {el-ph}} = \frac{1.04 + \mu ^* \ln {\big (\frac{\Theta ^{\rm{LT}}_{\rm{D}}}{1.45 T_{\rm{c}}}}\big )}{(1-0.62\mu ^*)\ln {\big (\frac{\Theta ^{\rm{LT}}_{\rm{D}}}{1.45 T_{\rm{c}}}}\big ) -1.04}, \end{aligned}$$

(7)

where $\mu ^* = 0.125$ (a value commonly accepted for d-electron elements) is the Coulomb pseudopotential of Morel and Anderson.^[43] The obtained electron-phonon coupling value is equal to 0.61(2) and it allows to classify the (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ system as a superconductor with intermediate electron-phonon coupling. Similar $\lambda _{\mathrm {el-ph}}$ values were reported for pure transition metals^[43] and various high entropy alloys based on transition metals.^[5,8,37] However, it should be noted that $\lambda _{\mathrm {el-ph}}$ = 0.61(2) is lower than 0.71(2) determined previously for another niobium- and titanium-rich HEA superconductor Nb$_{0.34}$Ti$_{0.33}$Zr$_{0.14}$Ta$_{0.11}$Hf$_{0.08}$.^[16] This finding may suggest that molybdenum and vanadium atoms have negative influence on the superconducting properties of the parent system.

Finally, by subtracting the phonon contribution from the total specific heat, the electron contribution can be calculated. The resulting data curve is shown in the upper inset of Figure 7 as $\ln ({C_{\rm{es}}/\gamma T_{\rm{c}}})$ as a function of $T^{-1}$. This dependence can be reproduced within the BCS theory of superconductivity using the formula^[44]:

$$\begin{aligned} C_{\rm{el}}(T) = A \gamma T_{\rm{c}} exp \big ( -\frac{\Delta _{\rm{0}}}{k_B T} \big ), \end{aligned}$$

(8)

where A is a constant and $\Delta _{\rm{0}}$ is a superconducting energy gap. A least-squares fit of this formula yields $\Delta _{\rm{0}}/k_B = 7.1(3) \,\rm{K}$ which gives a normalized energy gap $2\Delta _{\rm{0}}/k_B T_{\rm{c}} = 3.0(2)$. The obtained value is lower than 3.52 predicted by the BCS theory for a single and isotropic gap in the weak coupling limit,^[44] as well as lower than 3.7(1) determined for Nb$_{0.34}$Ti$_{0.33}$Zr$_{0.14}$Ta$_{0.11}$Hf$_{0.08}$.^[16]

3.5 Superconducting Phase Diagram

The values of the upper critical field $\mu _{\rm{0}}H_{\rm{c2}}$ as determined from measurements of magnetic susceptibility $\chi _{AC}$, mass magnetization $\sigma$, electrical resistivity $\rho ^{0}$ pct and specific heat $C_{\rm{p}}$ of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ are plotted as a function of temperature in Figure 8.

Fig. 8

Temperature dependence of the upper critical fields for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ derived from measurements of magnetic susceptibility, electrical resistivity and specific heat. The solid and dashed lines are fits of the WHH model (Eq. [9]), incorporating various effects, to the experimental points

To describe the experimental data, the Werthamer-Helfand-Hohenberg (WHH) formalism for isotropic-gap BCS superconductors in the dirty limit can be applied. Taking into account the spin-paramagnetic effect (described through the Maki parameter $\alpha_{\text{M}}$) and the spin-orbit scattering constant $\lambda _{\rm{SO}}$ one gets the formula^[45,46,47]:

$$\begin{aligned}{} & {} \ln \frac{1}{t}=\big (\frac{1}{2} +\frac{i\lambda _{\rm{SO}}}{4\gamma } \big ) \psi \big ( \frac{1}{2} + \frac{\overline{h}+\frac{1}{2}\lambda _{\rm{SO}} +i\gamma }{2t}\big ) \\{} & {} \quad + \big (\frac{1}{2} -\frac{i\lambda _{\rm{SO}}}{4\gamma } \big ) \psi \big ( \frac{1}{2} + \frac{\overline{h}+\frac{1}{2}\lambda _{\rm{SO}} -i\gamma }{2t}\big ) - \psi \big (\frac{1}{2}\big ), \end{aligned}$$

(9)

where $t=\frac{T}{T_{\rm{c}}}$, $\gamma \equiv \sqrt{(\alpha _{{\text{M}}} \overline{h})^2 -(\frac{1}{2} \lambda _{\rm{SO}})^2}$, $\overline{h}=\frac{4}{\pi ^2} \frac{H_{\rm{c2}}}{-dH_{\rm{c2}}(T)/dT}$ and $\psi$ is a digamma function. The solid line in Figure 8 is the theoretical curve simulated for Eq. [9], which was found to be the best fit to the experimental data. The obtained parameters are $\alpha _{\text{M}} = 1.3$, $\lambda _{\rm{SO}} = 1.2$ and $T_{\rm{c}} = 4.8(2)$ K, which results in the value of $\mu _{\rm{0}}H_{\rm{c2}}(0) = 6.8(1)$ T. The dashed lines in Figure 8 show the $\mu _{\rm{0}}H_{\rm{c2}}(T)$ dependence assuming the absence of the spin-orbit scattering effect ($\alpha _{\text{M}} = 1.3$ and $\lambda _{\rm{SO}} = 0.0$) and both spin-orbit scattering and spin-paramagnetic effects ($\alpha _{\text{M}} = 0.0$ and $\lambda _{\rm{SO}} = 0.0$).

In the next step, the Pauli limiting field $H_{\rm{P}}$ can be estimated based on the formula^[48]:

$$\begin{aligned} \mu _{\rm{0}}H_{\rm{P}} = 1.84 T_{\rm{c}}. \end{aligned}$$

(10)

Taking the $T_{\rm{c}}$ value derived from the fit of the WHH model, the obtained value is $\mu _{\rm{0}}H_{\rm{P}} = 8.8(4)$ T. Furthermore, using the relation^[47]:

$$\begin{aligned} \alpha _{{\text{M}}} =\sqrt{2} \frac{\mu _{\rm{0}}H_{\rm{c2}}^{\rm{orb}}}{\mu _{\rm{0}}H_{\rm{P}}}, \end{aligned}$$

(11)

one can calculate the orbital-limited upper critical field $\mu _{\rm{0}}H_{\rm{c2}}^{\rm{orb}}$, which is 8.1(4) T.

Table 1 summarizes the most important parameters describing the superconducting and normal states derived from the experimental data collected for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$. The determined values of the critical temperature and the upper critical field for (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ system are comparable to other HEA superconductors based on transition metals.^[5,7,8,37] However, the obtained $T_{\rm{c}}$ = 4.8(2) K and $\mu _{\rm{0}}H_{\rm{c2}}(T)$ = 6.8(1) T are much lower than $T_{\rm{c}}$ = 7.5(2) K and $\mu _{\rm{0}}H_{\rm{c2}}(T)$ = 12.2(1) T reported for Nb$_{0.34}$Ti$_{0.33}$Zr$_{0.14}$Ta$_{0.11}$Hf$_{0.08}$.^[16] Therefore, one tentatively conclude that in niobium- and titanium-rich HEA superconductors replacement of Zr and Ta atoms by Mo and V atoms leads to a deterioration of their superconducting properties.

Table 1 Summary of the Most Important Parameters Describing the Superconducting and Normal States of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$

Worth noting is the fact that in previously studied NbTa-based HEA superconductors,^[5,8,11] the upper critical field is limited by the orbital pair breaking effect ($\alpha _{\text{M}} <1$ and $H_{\rm{c2}}^{orb} < H_{\rm{P}}$). In the case of the studied (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ as well as Nb$_{34}$Ti$_{33}$Zr$_{14}$Ta$_{11}$Hf$_{8}$,^[16] $\alpha _{\text{M}} \approx$ 1.3 to 1.4, and thus $H_{\rm{c2}}^{orb} \approx H_{\rm{P}}$. This finding suggests that similarly to Nb–Ti alloy^[49] and Nb$_3$Sn^[50] superconductors, in niobium- and titanium-rich HEA superconductors, the upper critical field is not limited by relatively low $H_{\rm{c2}}^{orb}$. Therefore, it is plausible to assume that the superconducting properties of NbTi-based HEAs can be further improved by changing their chemical composition, i.e. by increasing Ti content.

3.6 Theoretical Modeling of the Electronic Structure

Total and partial atomic densities of states of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ calculated by the KKR-CPA method are presented in Figure 9. As one can notice, the total density of states (TDOS) shows a broad peak in the energy region close to the Fermi level $E_{\rm{F}}$. The maximum of the TDOS is located about 0.09 eV below $E_{\rm{F}}$, while the density of states at $E_{\rm{F}}$ is equal to $N^{\rm{KKR}}(E_{\rm{F}})$ = 1.85 st. eV$^{-1}$ f.u.$^{-1}$. In the case of partial atomic densities of states, it can be seen that in the investigated energy region, DOS maxima for Ti, V and Hf are located in the vicinity of $E_{\rm{F}}$, while DOS maxima for Nb and Mo are about 1.1 eV below the Fermi level.

Fig. 9

Electron density of states of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ calculated by the KKR-CPA method. Total DOS, plotted with gray solid line, and partial atomic densities, marked with colors and weighted by their atomic concentrations (Color figure online)

Taking $N^{\rm{KKR}}(E_{\rm{F}})$ value and using Eq. [6], the theoretical specific heat coefficient $\gamma _{th} ^{\rm{KKR}}$ = 4.36 mJ K$^{-2}$ mol$^{-1}$ is calculated. In the next step, the experimental value of $\gamma$ = 7.41 mJ K$^{-2}$ mol$^{-1}$ and $\gamma _{th} ^{\rm{KKR}}$ can be used to estimate the electron-phonon coupling constant $\lambda _{\mathrm{el-ph}} ^{\rm{KKR}} = \gamma / \gamma _{th} ^{\rm{KKR}} -1$ = 0.7.^[51,52] This value is slightly higher than experimentally determined $\lambda _{\mathrm {el-ph}}$ = 0.61(2). As a result, an overestimated $T_{\rm{c}}$ = 7.3 K is obtained from the McMillan’s formula (Eq. [7]) with $\mu ^* = 0.125$ and $\Theta _{\rm{D}}^{\rm{LT}} = 315(5) \,\rm{K}$. This finding indicates that the prediction of the critical temperature based on KKR-CPA and McMillan approximation is slightly overestimated, as seen previously for other HEA superconductors.^[8,11,52,53]

Total and partial atomic densities of states of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ calculated by the PAW method using VASP for three assumed random atomic arrangements are presented in the upper panel of Figure 10. As one can notice, the obtained results are very similar for all three random atomic arrangements. It suggests that in the case of HEAs, the influence of the local atomic configurations on their electronic structure in the energy region close to $E_{\rm{F}}$ is rather small or even negligible. This finding is in agreement with our previous results obtained by the PAW method for (NbTa)$_{0.67}$(MoHfW)$_{0.33}$^[8] and Nb$_{0.34}$Ti$_{0.33}$Zr$_{0.14}$Ta$_{0.11}$Hf$_{0.08}$.^[16] Therefore, the TDOS averaged over three considered atomic arrangements is calculated and shown together with the TDOS determined by KKR-CPA method in the bottom panel of Figure 10. A closer look at this graph reveals that the differences between the two curves are small. Thus, it can be said that the KKR-CPA and PAW methods produce consistent results. The computed density of states at $E_{\rm{F}}$ is equal to $N^{\rm{PAW}}(E_{\rm{F}})$ = 1.81 st. eV$^{-1}$ f.u.$^{-1}$. By adopting the same procedure which was employed for KKR-CPA data, one can calculate $\gamma _{th} ^{\rm{PAW}}$ = 4.27 mJ K$^{-2}$ mol$^{-1}$, $\lambda _{\mathrm{el-ph}} ^{\rm{PAW}}$ = 0.74 and $T_{\rm{c}}$ = 8.4 K. Again, the values of $\lambda _{\mathrm{el-ph}} ^{\rm{PAW}}$ and $T_{\rm{c}}$ are higher than experimentally determined parameters.

Fig. 10

Electron density of states of (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ calculated by the PAW method for three assumed random atomic arrangements which are presented in the right panel. TDOS averaged over three configurations is shown in the bottom panel together with corresponding KKR-CPA data

As it was mentioned above, in the case of many HEA superconductors, similar problems with reproducing the experimentally determined $\lambda _{\mathrm{el-ph}} ^{\rm{PAW}}$ and $T_{\rm{c}}$ using DFT calculations have been reported.^{[8,11,16,52,53]} A most plausible explanation for this fact would seem to be that the electronic structure calculations assume the absence of any crystallographic defects (vacancies, dislocations, grain boundaries), while the real polycrystalline HEA samples prepared using arc-melting method contains a relatively large number of these types of defects. The influence of crystallographic defects on the superconducting properties of various superconductors has recently been reported in works.^{[49,54,55,56]} Therefore, the polycrystalline nature and the high degree of structural disorder in the studied (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ sample are probably mainly responsible for discrepancies between theoretical predictions and experimental data.

4 Conclusions

Structural and physical properties of a new niobium- and titanium-rich high-entropy alloy (NbTi)$_{0.67}$(MoHfV)$_{0.33}$ were determined using various bulk physical properties measurements. SEM and XRD data reveal that the alloy consists of two bcc phases. The chemical composition of the major phase (91 wt pct) is almost the same as the nominal composition, while the secondary phase contains a slightly higher Mo content. The electrical resistivity, magnetic and specific heat data show that the studied material is the type-II superconductor with $T_{\rm{c}} \approx$ 4.8 K, the intermediate electron-phonon coupling $\lambda _{\mathrm {el-ph}}$ = 0.61(2) and the upper critical field $\mu _{\rm{0}} H_{\rm{c2}}(0) \approx$ 6.8 T. A comparison of these parameters with $T_{\rm{c}} \approx$ 7.5 K and $H_{\rm{c2}}(0) \approx$ 12.2 T observed for another niobium- and titanium-rich HEA superconductor Nb$_{0.34}$Ti$_{0.33}$Zr$_{0.14}$Ta$_{0.11}$Hf$_{0.08}$ leads to the conclusion that molybdenum and vanadium atoms have a negative influence on the superconducting properties of the parent system. However, in contrast to NbTa-based superconductors, the upper critical field in NbTi-based HEAs is not limited by relatively low $H_{\rm{c2}}^{orb}$. This finding suggests that the new family of NbTi-based HEA superconductors may have great application potential in superconducting magnets production and their superconducting properties can be further improved by changing chemical composition, i.e. by increasing Ti content. The electronic structure calculations performed using KKR-CPA and PAW methods give very consistent results. However, as it is shown, the values of $\lambda _{\mathrm {el-ph}}$ and $T_{\rm{c}}$ obtained using the calculated density of states data for the studied system are slightly overestimated compared to the experimental data.