# Superconducting Ti_{15}Zr_{15}Nb_{35}Ta_{35} High-Entropy Alloy With Intermediate Electron-Phonon Coupling

^{1}State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing, China^{2}Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, China^{3}Songshan Lake Materials Laboratory, Dongguan, China^{4}High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei, Anhui, China^{5}Laboratory for Microstructures, Institute of Materials, Shanghai University, Shanghai, China

The body-centered cubic (BCC) Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 K^{2} and β = 0.28 ± 0.01 mJ/mol K^{4}, 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 (Δ*C*/γ*T*_{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 theTi_{15}Zr_{15}Nb_{35}Ta_{35} 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 Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} HEA with a SC transition temperature *T*_{c} (*T*_{c} ≈ 8 K). Our analysis indicates that the Ti_{15}Zr_{15}Nb_{35}Ta_{35} HEA is an intermediate electron-phonon coupled superconductor, which naturally explains its higher *T*_{c} in comparison with Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} (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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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} = ∑_{i}*c*_{i}*a*_{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.

**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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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**. Temperature dependence of electrical resistivity ρ(*T*) in zero field of the Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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**. *C*(*T*)of Ti_{15}Zr_{15}Nb_{35}Ta_{35}for *H* = 0, 4, 8 and 10 T. The upper inset shows *C*/*T* vs *T*^{2} under magnetic field. And the lower right inset is *C*_{es}/γ*T*_{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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} HEA. The susceptibility becomes negative (diamagnetic) below temperature 8.5 K, with a strong diamagnetic response due to the Meissner effect. Here, the measured χ_{0} was corrected for demagnetization fieldsχ = χ_{0}/(1−*Dχ*_{0}), 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**. Temperature dependence of magnetic susceptibility χ(*T*) measured under zero-field-cooled (ZFC) and field-cooled (FC) conditions at 3 mT for Ti_{15}Zr_{15}Nb_{35}Ta_{35} samples. The left inset shows isothermal magnetization *M*(*H*)curves for Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 field*H*_{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) = 1.86*T*_{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):

where *t* = *T*/*T*_{c} and $\stackrel{\u0304}{h}=(4/{\pi}^{2})\left[{H}_{c2}(T)/{|d{H}_{c2}/dT|}_{{T}_{c}}\right]$. 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 Φ_{0} is the quantum flux (= *h*/2*e* = 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\text{}\kappa $, and hence the Ginzburg-Landau parameter, κ = 17.3.

**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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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*) = γ*T*+*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. *T*^{2} 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+β*T*^{2} yields the intercept γ = 9.3 ± 0.1 mJ/mol K^{2}, the slope β = 0.28 ± 0.01 mJ/mol K^{4} 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 Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} HEA.

To elucidate the nature of the SC state of theTi_{15}Zr_{15}Nb_{35}Ta_{35} 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., Δ*C*(*T*) = *C*(*T*)−*C*_{lattice}(*T*)−γ*T*. The resultant Δ*C*(*T*)/*T* at *H* = 0 is shown in Figure 6. Utilizing the balance of entropy around the transition, Δ*C*/γ*T*_{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 (Δ*C*/γ*T*_{c} = 1.43).

Based on the BCS theory (Tinkham, 2004), the energy required to break up a Cooper pair is about 2Δ_{0}, and the number of Cooper pairs broken up is proportional to exp(−2Δ(0)/*k*_{B}*T*), 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}/γ*T*_{c} = *A*exp(−*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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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}/γ*T*_{c} = 10.1 × exp(-1.66*T*_{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 Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} HEA (Koželj et al., 2014).

It is found that the measured SC properties in Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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):

where *k*_{B}is 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},2Δ_{0} and λ were determined by Equations (2–4) based on the experimentally measuredΔ*C*/γ*T*_{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 2Δ_{0}/*k*_{B}*T*_{c} is 3.52 for the weak-coupling BCS conventional superconductors. In our HEA, 2Δ_{0}/*k*_{B}*T*_{c}is calculate to be 3.72 ± 0.02 using the energy gap 2Δ_{0} of 2.57 ± 0.01 meV, larger than the reported Ta _{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} HEA of 3.50 indicating the stronger electron-phonon coupling (Takeya et al., 2005). Therefore, these values suggest that the Ti_{15}Zr_{15}Nb_{35}Ta_{35} HEA is an intermediate-coupled superconductor, when compared with the reported weak-coupled Ta _{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} HEA.

**Table 2**. SC parameters of Ti_{15}Zr_{15}Nb_{35}Ta_{35} HEA and the reported Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} HEA (Koželj et al., 2014).

The electronic-phonon coupling parameter λ in the reported weak-coupling Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11} 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 theTi_{15}Zr_{15}Nb_{35}Ta_{35} HEA and the reported Ta_{34}Nb_{33}Hf_{8}Zr_{14}Ti_{11}HEA, respectively. Therefore, the improvement of SC temperature *T*_{c} is related to the increase of the electron density of state at the Fermi level*N*(*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 Ti_{15}Zr_{15}Nb_{35}Ta_{35} 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 Ti_{15}Zr_{15}Nb_{35}Ta_{35}HEA 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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Keywords: high-entropy alloys, superconducting, BCS-type, specific heat jump, intermediate electron-phonon coupled

Citation: Yuan Y, Wu Y, Luo H, Wang Z, Liang X, Yang Z, Wang H, Liu X and Lu Z (2018) Superconducting Ti_{15}Zr_{15}Nb_{35}Ta_{35} High-Entropy Alloy With Intermediate Electron-Phonon Coupling. *Front. Mater.* 5:72. doi: 10.3389/fmats.2018.00072

Received: 04 July 2018; Accepted: 09 November 2018;

Published: 29 November 2018.

Edited by:

John L. Provis, University of Sheffield, United KingdomReviewed by:

Jacqueline Krim, North Carolina State University, United StatesAlexander Petrovic, Nanyang Technological University, Singapore

Copyright © 2018 Yuan, Wu, Luo, Wang, Liang, Yang, Wang, Liu and Lu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yuan Wu, wuyuan@ustb.edu.cn

Zhaoping Lu, luzp@ustb.edu.cn