Superconducting Ti15Zr15Nb35Ta35 High-Entropy Alloy With Intermediate Electron-Phonon Coupling

The body-centered cubic (BCC) Ti₁₅Zr₁₅Nb₃₅Ta₃₅ high-entropy alloy showed superconducting behavior at around 8 K. The electronic specific heat coefficient γ and the lattice specific heat coefficient β were determined to be γ = 9.3 ± 0.1 mJ/mol K² and β = 0.28 ± 0.01 mJ/mol K⁴, respectively. It was found that the electronic specific heat C_es does follow the exponential behavior of the Bardeen-Cooper-Schrieffer (BCS) theory. Nevertheless, the specific heat jump (ΔC/γT_c) at the superconducting transition temperature which was determined to be 1.71 deviates appreciably from that for a weak electron-phonon coupling BCS superconductor. Within the framework of the strong-coupled theory, our analysis suggests that theTi₁₅Zr₁₅Nb₃₅Ta₃₅ HEA is an intermediate electron-phonon coupled BCS-type superconductor.

Introduction

Within the past decade, high-entropy alloys (HEAs) have attracted extensive attention due to their unique compositions, interesting microstructures, and promising properties. Traditional alloys usually contain one element as the principle constituent with some other minor elements incorporated for property optimization (Lu et al., 2015). However, HEAs contain at least four principle elements in equal or near-equal atomic ratio (Li et al., 2017). It is interesting to note that, despite containing a large number of components, HEAs tend to form simple face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) structure instead of complex phases and intermetallic compounds (Urban and Feuerbacher, 2004; Conrad et al., 2009). Most current studies are focusing on the relationship between microstructure and mechanical properties (Yeh et al., 2004; Yeh, 2006; Zhou et al., 2007; Wen et al., 2009; Senkov et al., 2010, 2011; Chuang et al., 2011; Hsu et al., 2011), but limited work on the physical properties. Recently, it was revealed that the Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA became a superconductor at 7.3 K in a weak electron-phonon coupled matter (Koželj et al., 2014), which attracted tremendous interests. Recently, it was reported that (TaNb)_0.67(HfZrTi)_0.33 HEA even showed a robust superconducting (SC) behavior under extreme pressure up to 160 GPa, a pressure like that within the outer core of the earth (Guo et al., 2017). Note that Ti, Zr, Nb, and Ta are known to be SC elements under ambient pressure, and proper mixing of these elements is favorable for a formation of a single solid solution phase (Todai et al., 2017). Therefore, careful studies on the SC behavior of the Ti-Zr-Nb-Ta HEA system is interesting, which may offer new clues for understanding SC mechanism of HEAs. Herein, we report synthesis and characterization of a novel four-component Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA with a SC transition temperature T_c (T_c ≈ 8 K). Our analysis indicates that the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA is an intermediate electron-phonon coupled superconductor, which naturally explains its higher T_c in comparison with Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA compound.

Materials and Methods

99.9% pure Ti, Zr, Nb, and Ta were used as starting materials for the preparation of alloy ingots with a nominal composition of Ti₁₅Zr₁₅Nb₃₅Ta₃₅ (at. %). The materials were put in a water-cooled copper crucible in a vacuum arc furnace. Then the materials were melted by an argon arc plasma flame at half atmospheric pressure under argon atmosphere. A dc power source with about 100 A was adjusted for 10 s and then increased to 300–400 A for about 1 min. The arc-flame between the tungsten electrode and the surface of the molten sample was maintained at about 20 mm during the arc melting process. The ingots were re-melted at least six times to ensure chemical homogeneity and subsequently drop-cast into a copper mold with a dimension of Φ10 × 60 mm. Phase constituents were characterized by X-ray diffraction (XRD) using Cu-Kα radiation. Specific heat, resistivity and magnetization measurements were carried out on a physical property measurement system (PPMS) from Quantum Design Company.

Results and Discussion

X-ray diffraction pattern of the as-cast Ti₁₅Zr₁₅Nb₃₅Ta₃₅ sample is shown in Figure 1 in which all diffraction peaks can be indexed to a BCC lattice, indicating formation of a single crystalline phase in the current alloy. The lattice constant of the current HEA using Vegard's rule of mixture(Vegard, 1921), a_mix = ∑_ic_ia_i, which is valid for completely random mixing of the elements, was calculated to be a = 3.340 Å. Here c_i is atomic fraction and a_i is lattice parameter of the element i (Table 1) (Koželj et al., 2014). The calculated value is in good agreement with experimental value (i.e., a = 3.329 Å, suggesting that the constituent elements are likely randomly mixed in the current BCC lattice.

FIGURE 1

Figure 1. X-ray diffraction pattern of the as-cast Ti₁₅Zr₁₅Nb₃₅Ta₃₅ sample.

TABLE 1

Table 1. Lattice constant a (Koželj et al., 2014), SC transition temperature T_c (Ashcroft and Mermin, 1976) and the specific heat coefficients γ (Tari, 2003) of pure constituent elements and the investigated Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA.

Temperature dependence of electrical resistivity ρ (T) in zero field is shown in Figure 2, and the inset shows the variation of resistivity as a function of temperature under magnetic field ρ(T)H up to 10 T field. The resistivity value is 25 ± 1 μΩ cm at room temperature, and decreases with temperature as expected for metallic behavior(Awana et al., 2010). At 8 K, the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ sample exhibits SC behavior, as the resistivity gradually approaches zero. Theρ (T)H measurements reveal that the SC transition temperature is shifted to lower temperatures as the applied magnetic field is increases. Figure 3 shows C (T) of the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA in the magnetic field of 0, 4, 8, and10 T. An anomaly at about 8 K associated with the SC transition is visible in the magnetic field of 0 and 4 T, but disappear in the magnetic field of 10 T. This observation is in good agreement with the results of electrical resistivity ρ (T) (shown in Figure 2).

FIGURE 2

Figure 2. Temperature dependence of electrical resistivity ρ(T) in zero field of the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA. The inset shows the variation of resistivity as a function of temperature under magnetic field ρ (T)H up to10 T field.

FIGURE 3

Figure 3. C(T)of Ti₁₅Zr₁₅Nb₃₅Ta₃₅for H = 0, 4, 8 and 10 T. The upper inset shows C/T vs T² under magnetic field. And the lower right inset is C_es/γT_c plotted on a logarithmic scale as a function of T_c/T.

Figure 4 shows temperature dependence of magnetic susceptibility χ (T) under zero-field-cooled (ZFC) and field-cooled (FC) conditions at 3 mT for the current Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA. The susceptibility becomes negative (diamagnetic) below temperature 8.5 K, with a strong diamagnetic response due to the Meissner effect. Here, the measured χ₀ was corrected for demagnetization fieldsχ = χ₀/(1−Dχ₀), where χ is the susceptibility corrected for demagnetization, and D is the demagnetization factor. Exact value of D was obtainable only for ellipsoids. The ratio of length to diameter for our cylinder-like sample is about 0.6, then the average value of D for uniformly magnetized cylinders is 0.43 according to the results computed by Brown (1960) and Crabtree (1977). Thus, it can also be clearly observed that the value of magnetic susceptibility (χ) is about −1, corrected for the demagnetization factor. This observation indicates that the SC volume fraction is almost 100%. The left inset of Figure 4 shows isothermal magnetization M(H) curves at 3–10 K in the field range up to 80 kA/m. A linear response due to Meissner effect is observed with a slope nearly −1, which confirms the bulk superconductivity. Then the magnitude of M decreases with increasing H, which is typical in the vortex state of a type-II superconductor (Awana et al., 2010). Based on the method reported in Ref. (Abdel-Hafiez et al., 2013), the lower critical field (H_c1) was obtained by measuring the virgin M(H) curve at various temperatures, where the regression coefficient function R(H) of a linear fit to the data between 0 and H departs from 1. Then the plot of H_c1 (T) is shown in the lower right inset of Figure 4. By linearly extrapolating H_c1(T) to zero temperature, we obtain H_c1 (0) = 55 mT. It should be noted that this is a very rough approximation in the absence of data at lower T.

FIGURE 4

Figure 4. Temperature dependence of magnetic susceptibility χ(T) measured under zero-field-cooled (ZFC) and field-cooled (FC) conditions at 3 mT for Ti₁₅Zr₁₅Nb₃₅Ta₃₅ samples. The left inset shows isothermal magnetization M(H)curves for Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA at 3-10 K in the field range up to 80 kA/m. The lower right inset is the temperature dependence of the lower critical fieldH_c1(T).

The upper critical field H_c2 corresponding to the temperatures where the resistivity drops to 90% of the normal state resistivity ρ(T, H) (see inset of Figure 2) at the onset critical temperature $T_c^{onset}$ determined from the intersection of the two extrapolated lines of ρ (T) in the applied magnetic fields (Jha et al., 2014). The upper critical field H_c2 determined from resistivity (black circles) and from specific heat measurements (red squares) are shown in Figure 5. For a single-band superconductor in the dirty limit (Werthamer and Helfand, 1966; Wang et al., 2015), the orbital limit is given by $H_{c2}^{orb}(0)=-0.69{d{H}_{c2}/dT|}_{T=T_C}{T}_C$. Alternatively, the Pauli-limiting field for a weakly coupled BCS superconductor in the absence of spin-orbit scattering can be estimated as (Clogston, 1962) H_P(0) = 1.86T_c. The Maki parameter α is defined as $\alpha \mathrm{\text{=}}\sqrt{2}{H}_{c2}^{orb}(0)/H_P(0)$. Assuming T_C = 8.5 K, and we have d_{Hc2/dT|T = TC} = −2 T/K from Figure 5. Then α is calculated to be 1.05. We used the full Werthamer-Helfand-Hohenberg (WHH) formula that incorporates the spin-paramagnetic effect via the Maki parameter α and the spin-orbit scattering constant λ_so to describe the experimental H_c2 data (Werthamer and Helfand, 1966):

$\begin{array}{l}\ln \frac{1}{t}=\\ {\displaystyle \sum _{\nu =-\infty }^{\infty }\left\{\frac{1}{\left|2\nu +1\right|}-{\left[\left|2\nu +1\right|+\frac{\overline{h}}{t}+\frac{{\left(\alpha \overline{h}/t\right)}^2}{\left|2\nu +1\right|+\left(\overline{h}+{\lambda }_{so}\right)/t}\right]}^{-1}\right\}}& (1)\end{array}$

where t = T/T_c and $\bar{h}=(4/{\pi }^2)\left[H_{c2}(T)/{|d{H}_{c2}/dT|}_{T_c}\right]$. As shown in Figure 5, the best fit can reproduce the experimental H_c2 very well. By fixing α = 1.05 to fit the H_c2 data, we have λ_so = 23.2, H_c2 (0) = 11.6 T. With these results for H_c1 (0) and H_c2 (0), we can calculate several SC parameters for our HEA. From $H_{c2}\mathrm{\text{=}}{\Phi }_0/2\pi {\xi }_{GL}^2$, where Φ₀ is the quantum flux (= h/2e = 2.07 × 10⁻¹⁵ Wb), we find a Ginzburg-Landau coherence length ξ_GL(0) = 53.3 Å = 5.33 × 10⁻⁹ m. Assuming κ = λ_GL/ξ_GL is temperature independent, then λ_GL(0) = κ_ξGL(0). Knowingξ_GL(0), H_c1(0), and $H_{c1}(0)\mathrm{\text{=}}({\Phi }_0/4\pi {(\kappa *{\xi }_{GL}(0))}^2)ln\kappa$, and hence the Ginzburg-Landau parameter, κ = 17.3.

FIGURE 5

Figure 5. The upper critical field H_c2 from resistivity (black circles) and from specific heat measurements (red squares) as a function of temperature for the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA. The blue and pink lines are the WHH predictions with the parameter α = 0, λ_so = 0, and α = 1.05, λ_so = 23.2, respectively.

With respect to the superconductivity in Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA, the most interesting part is the detailed analysis of the anomaly in the specific heat C(T). The low-temperature specific heat in the normal state is expressed as C(T) = γT+C_lattic(T), where the electronic specific heat coefficient γ measures the degree of electronic contribution and $C_{lattic}(T)=\beta T^3+\sigma T^5$ represents the phonon contribution in which β is the lattice specific heat coefficient. Then the plot of C/T vs. T² is shown in the upper inset of Figure 3. The analysis of the normal-state specific heat (in 8 or 10 T field) with the expression C/T = γ+βT² yields the intercept γ = 9.3 ± 0.1 mJ/mol K², the slope β = 0.28 ± 0.01 mJ/mol K⁴ and the Debye temperature ${\theta }_D={(12{\pi }^{4}R/5\beta )}^{1/3}=$191 ± 3 K. Clearly, for the current HEA, the electronic specific heat coefficient γ increases by about 12% as compared with that of the reported Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA.

To elucidate the nature of the SC state of theTi₁₅Zr₁₅Nb₃₅Ta₃₅ HEA, e.g., Bardeen-Cooper-Schrieffer (BCS) type (Bardeen et al., 1957) or unconventional, it is of interest to calculate the discontinuous jump of specific heat at the SC transition temperature, i.e., ΔC(T) = C(T)−C_lattice(T)−γT. The resultant ΔC(T)/T at H = 0 is shown in Figure 6. Utilizing the balance of entropy around the transition, ΔC/γT_c (the specific-heat jump) is 1.71 ± 0.02 for the current HEA which deviates appreciably from that for a weak electron-phonon coupling BCS superconductor (ΔC/γT_c = 1.43).

FIGURE 6

Figure 6. ΔC(T)/Tvs. T at H = 0 for theTi₁₅Zr₁₅Nb₃₅Ta₃₅ HEA.

Based on the BCS theory (Tinkham, 2004), the energy required to break up a Cooper pair is about 2Δ₀, and the number of Cooper pairs broken up is proportional to exp(−2Δ(0)/k_BT), which leads an exponential temperature dependence of the specific heat at sufficiently low temperature. The electronic specific heat C_es(T) [C_es(T) = C(T, H)−C_lattice(T)] in the BCS theory is expressed by C_es/γT_c = Aexp(−BT_c/T) (Tari, 2003), where A and B are two constants whose values relate to the temperature interval used. Evidently in the lower inset of Figure 3, C_es(T)of the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA reflects clearly an exponential behavior. The fit C_es(T) of the current HEA at 2.5 < T_C/T < 6 yields C_es/γT_c = 10.1 × exp(-1.66T_c/T). Evidently, it does have the characteristics of exponential behavior predicted by the BCS theory, but the factors are far from the weak electron-phonon coupling (Takeya et al., 2005). This phenomenon is different from the reported Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA (Koželj et al., 2014).

It is found that the measured SC properties in Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA agree extremely well with the following strong electron-phonon coupling corrections (valid for T_c/ω_log < < 1, where ω_log is a characteristic phonon temperature; McMillan, 1968; Ishikawa and Toth, 1971; Allen and Dynes, 1975; Carbotte, 1990):

$\begin{array}{rc}{\left[\Delta C/\gamma T_c\right]}_{T_c}=1.43\left[1+53{(T_c/{\omega }_{log})}^{2}ln({\omega }_{log}/3T_c)\right]& (2)\end{array}$

$\begin{array}{rc}2{\Delta }_0/k_B{T}_c=3.53\left[1+12.5{(T_c/{\omega }_{log})}^{2}ln({\omega }_{log}/2T_c)\right]& (3)\end{array}$

$\begin{array}{rc}{T}_c=\frac{{\omega }_{log}}{1.2}exp(-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{*}(1+0.62\lambda )})& (4)\end{array}$

where k_Bis the Boltzmann constant, λ the electronic-phonon coupling parameter, and μ* the Coulomb pseudopotential taken to be 0.13 here, which is an average value used commonly for intermetallic superconductors (Klimczuk et al., 2012; von Rohr et al., 2016). The values of ω_log,2Δ₀ and λ were determined by Equations (2–4) based on the experimentally measuredΔC/γT_c, as compiled in Table 2. It is apparent that the calculated results are reasonably satisfied with the condition T_c/ω_log < < 1. The value of 2Δ₀/k_BT_c is 3.52 for the weak-coupling BCS conventional superconductors. In our HEA, 2Δ₀/k_BT_cis calculate to be 3.72 ± 0.02 using the energy gap 2Δ₀ of 2.57 ± 0.01 meV, larger than the reported Ta ₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA of 3.50 indicating the stronger electron-phonon coupling (Takeya et al., 2005). Therefore, these values suggest that the Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA is an intermediate-coupled superconductor, when compared with the reported weak-coupled Ta ₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA.

TABLE 2

Table 2. SC parameters of Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA and the reported Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA (Koželj et al., 2014).

The electronic-phonon coupling parameter λ in the reported weak-coupling Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁ HEA is calculated to be 0.66 ± 0.01 using the equation $T_c=({\theta }_D/1.45)exp\left[-(1+\lambda )/(\lambda -{\mu }^{*})\right]$ (Takeya et al., 2005). Clearly this value of λ is lower than that of the current HEA (0.84 ± 0.02). The electron density of state at the Fermi level N(E_F) is estimated to be 1.07 ± 0.01 and 1.06 ± 0.02 by $\gamma =\frac{2}{3}{\pi }^2{k}_B^2(1+\lambda )N(E_F)$for theTi₁₅Zr₁₅Nb₃₅Ta₃₅ HEA and the reported Ta₃₄Nb₃₃Hf₈Zr₁₄Ti₁₁HEA, respectively. Therefore, the improvement of SC temperature T_c is related to the increase of the electron density of state at the Fermi levelN(E_F). In addition, the electron-phonon coupling is enhanced, which also might contribute to the increase of T_c. It was also found in the current HEA that the SC properties, e.g., T_c and γ ,do not obey the Vegard's rule of mixture (Vegard, 1921), $p_{mix}=\sum _{i=1}^n{c}_i{p}_i$, where c_i and p_i are the atomic percentage and the SC property of each constituent element (Table 1), respectively. This observation suggests that electronic properties are not a “cocktail” of properties of the constituent elements. Theoretical description of the electronic properties of HEAs is thus a highly complex problem.

Conclusions

The Ti₁₅Zr₁₅Nb₃₅Ta₃₅ HEA with a single BCC lattice was synthesized and found to be a type-II superconductor. The analysis of all the specific heat data shows that this new Ti₁₅Zr₁₅Nb₃₅Ta₃₅HEA is a BCS-type phonon-mediated superconductor in the intermediate electron-phonon coupling regime, which is different from the previous report. Despite the complex electronic properties, our results also indicate that the characteristics of superconductivity of HEAs can be tailored by varying their chemical composition. Further experiments on its transport properties under high pressure would be very interesting to check the robustness of superconductivity.

Author Contributions

YY performed the experiments. YW, HL, ZW, XL, ZY, HW, XL, and ZL supervised the project. YY, HL, and ZW analyzed the data. YY, YW, and ZL wrote the paper. All authors commented on the manuscript.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This research was supported by National Natural Science Foundation of China (Nos. 11790293, 51871016, 51671018, 51671021, 51531001 11822411, 11674372, 11704385 and 11874359), 111 Project (B07003), International S&T Cooperation Program of China (2015DFG52600), Program for Changjiang Scholars and Innovative Research Team in University of China (IRT_14R05), and the Projects of SKLAMM-USTB (2016Z-04, 2016-09, 2016Z-16). YW acknowledges the financial support from the Top-Notch Young Talents Program and Fundamental Research Fund for the Central Universities. HL acknowledges the financial support from the Chinese Academy of Sciences (SPRP-B: XDB25000000 and the Youth Innovation Promotion Association).

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