Superconductivity in HEA-type compounds

Since the discovery of superconductivity in a high-entropy alloy (HEA) Ti-Zr-Nb-Hf-Ta in 2014, the community of superconductor science has explored new HEA superconductors to find the merit of the HEA states on superconducting properties. Since 2018, we have developed HEA-type compounds as superconductors or thermoelectric materials. As well known, compounds like intermetallic compounds or layered compounds are composed of multi crystallographic sites. In a HEA-type compounds, one or more sites are alloyed and total mixing entropy satisfies with the criterion of HEA. Herein, we summarize the synthesis methods, the crystal structural variation and superconducting properties of the HEA-type compounds, which include NaCl-type metal tellurides, CuAl2-type transition metal zirconides, high-Tc cuprates, and BiS2-based layered superconductors. The effects of the introduction of a HEA site in various kinds of complicated compounds are discussed from the structural-dimensionality viewpoint.

unconventional superconductors over the last three decades, recent works on new superconductors have focused on various kinds of materials, which includes complicated compounds and pure metals as well. Among them, high-entropy-alloy (HEA) superconductors have been a developing field of study [15].
HEA is an alloy possessing high configurational mixing entropy (ΔSmix), which is achieved by making the alloy with more than five constituent elements with an occupancy ranging 5 to 35 at% for each element [16,17]. Typically, ΔSmix of HEA is calculated by ΔSmix = -R Σi ci ln ci, where ci and R are the compositional ratio and the gas constant [17], and reaches 1.5R. Due to high ΔSmix, HEAs exhibit stability or high performance in high temperature and/or extreme conditions [17]. Therefore, HEAs have been extensively studied in the fields of materials science and engineering.
In 2014, Koželj et al. reported superconductivity with Tc = 7.3 K in a HEA Ta0.34Nb0.33Hf0.08Zr0.14Ti0.11 [18]. The HEA superconductor has a bcc structure with a space group of Im-3m. In Fig. 1, we compare the crystal structures of (a) a pure Nb metal (Tc = 9.2 K), (b) a NbTi alloy (Tc ~ 10 K), which is the mostly-used practical superconductor, and (c) HEA Ta0.34Nb0.33Hf0.08Zr0.14Ti0.11. All those materials show superconductivity and the crystal structure type is the same. The difference between them is the mixing entropy ΔSmix and Tc. Although the Tc of HEA is lower than that of the other two, it was surprising for researchers that such a disordered alloy exhibits superconductivity with bulk nature. After the discovery of Ref. 18, various HEA superconductors have been developed; material information [18][19][20][21][22][23][24][25][26] is listed in Table I. As shown in Fig. 2, a superconducting transition was observed in Ta0.34Nb0.33Hf0.08Zr0.14Ti0.11 [18]. The temperature dependence of electrical resistivity ( Fig. 2(a)) shows metallic behavior but exhibits a relatively small residual resistivity ratio (RRR) at low temperatures. This would be due to the presence of disorder, and a similar trend has been observed in pressure. As reported in Ref. 20, the Tc of Ta-Nb-Hf-Zr-Ti slightly increases by external pressure effect and the 10 K-class Tc maintains under extreme pressures like 200 GPa. However, the robustness of superconductivity to extremely high pressure was reported for simpler NbTi with a clear increase in Tc to 19.1 K at 261 GPa [28]. Therefore, the issue if HEA can improve the stability of superconductor under extremely high pressures has not been clarified.

1-3. Concept of HEA-type compounds
As described in subsection 1-2, superconductivity in HEAs has been discovered and been regarded as a new research field of superconductivity. However, the merit of HEA states for superconductors has not been fully understood. Therefore, development of new types of HEA superconductors is needed. The hint to expand the material variation of HEA superconductors was proposed in Ref. 29, in which a HEA superconductor with a CsCl-type structure was reported. Since the CsCl structure contains two independent crystallographic sites, we have flexibility of elemental solution at the two sites. When calculating total ΔSmix of (ScZrNbTa)0.65(RhPd)0.35 by taking the sum of ΔSmixs for site-1 and site-2, it appears to reach very high ΔSmix of 1.79R. A similar site separation has been observed in (Nb0.11Re0.56)(HfZrTi)0.33 [30]. Motivated by those studies on HEAs with site separation, we have tried to synthesize various "HEA-type compounds", which contain NaCl-type metal chalcogenides [31][32][33], CuAl2-type tetragonal TrZr2 (Tr: Fe, Co, Ni, Cu, Rh, Ir) [34,35], high-Tc RE123 cuprates [36], and BiS2-based layered superconductors [37,38]. The concept of HEA-type compounds it that we achieve a high ΔSmix by site-selective alloying. As shown in Fig. 4, HEA-type compounds have a HEA-type site, in which five or more elements are solving and a normal site, which is not in the HEA state. The list of superconducting HEA-type compounds is shown in Table II. By studying HEA effects to crystal structure and physical properties in various crystal structures, we could identify the merit of HEA states in those compounds. In section 2, we review the material synthesis, crystal structure, and physical properties of newly synthesized HEA-type compounds.  Motivated by these facts, we tried to synthesize HEA-type tellurides MTe where the M site is in the HEA state (see Fig. 4(b) for crystal structure) by high-pressure synthesis.
Figure 5(a) shows the temperature dependence of electrical resistivity for AgInSnPbBiTe5, in which the M site is evenly occupied by Ag, In, Sn, Pb, and Bi (five metals) [31]. Very small RRR was observed, which is a similar trend to that in HEA superconductors [18]. In addition, four different MTe (M: Ag, In, Cd, Sn, Sb, Pb, Bi) superconductors with a HEA-type site has been obtained [32].
Interestingly, there is a correlation between the lattice constant and Tc in HEA-type MTe. In Figure   5(b), the data for typical MTe superconductors are plotted. It is found that the trend that Tc increases with increasing lattice constant is common among the plotted MTe. The Tcs of HEA-type are, however, lower than those of low-entropy tellurides, such as InTe and (In,Sn)Te. Therefore, the introduction of the HEA-type M site is found to negatively work for Tc in MTe. This negative effect would be due to the direct effect of strong disorder to the M-Te bonding states and hence electronic states.

2-2. Hybrid high-entropy alloying in MCh
The Te site of MTe can be substituted by S and Se. The flexibility of both M and Te sites to element substitution enables us to design "hybrid HEA", in which both sites are alloyed [33]. Figure   6(a) shows the X-ray diffraction patterns for (Ag0.25In0.25Pb0.25Bi0.25)Te1-xSex. Notably, mixing many elements at two sites does not result in phase separation, and a single-phase sample was obtained for x = 0.25, while small impurity phases were detected for x = 0.5. For x = 0.25, as displayed in Fig. 6(b), the ΔSmix for the M and Ch sites are 1.38R and 0.51R, and the total ΔSmix reaches 1.89R. Furthermore, the total ΔSmix for x = 0.5 reaches 2.00R. These ΔSmix values are clearly higher than that for HEAs (Table I). Therefore, alloying at two or more sites (hybrid high-entropy alloying) results in very high total ΔSmix.
A superconducting transition with Tc > 2 K was observed for x = 0 and 0.25. By focusing on these two phases, interesting trend was found. Although the Tc for x = 0.25 is lower than that for x = 0, the suppression of Tc for x = 0.25 under magnetic fields is clearly smaller than that for x = 0. From the estimation of upper critical field (Hc2), it was found that the Hc2 (0 K) for x = 0.25 is higher than that for x = 0. In addition, from the measurements of the magnetization-field loop, it was confirmed that the critical current density (Jc) at 1.8 K for x = 0.25 is larger than that for x = 0. These results suggest that an increase in ΔSmix may be useful to improve Hc2 and/or Jc characteristics of superconductors if the problem on the suppression of Tc could be solved.

Cuprate (high-Tc) superconductors REBa2Cu3O7-d
As mentioned in introduction, cuprates (Cu oxides) have been extensively studied in the fields of science and engineering because of its high Tc. Among them, REBa2Cu3O7-d (RE123) system [5] is one of practical materials for superconductivity application. In addition, a high Jc was reported in RE123 samples with three elements at the RE site [48]. Motivated by the fact, we synthesized polycrystalline samples of REBa2Cu3O7-d with different ΔSmix for the RE site [36] by standard solidstate reaction in air. In the study, two-step annealing was performed to optimize oxygen content (d) because oxygen content affect crystal structure (orthorhombicity) and superconducting properties of REBa2Cu3O7-d. A high superconducting property is generally achieved in an orthorhombic phase in the system, we estimated the orthorhombicity parameter (OP), which is given by 2|a-b|/(a+b), and plotted the estimated Tc and magnetic Jc (T = 2 K, B = 1 T) for REBa2Cu3O7-d as a function of OP as shown in Fig. 8. Note that the data points are colored according to the number of elements contained at the RE site. From the plot, it is found that Tc does not show a remarkable correlation with ΔSmix (RE site) and exhibits a clear correlation with OP. Jc also exhibits a trend of improvement with increasing OP. These facts suggest that disorder introduced by high-entropy alloying at the RE site (see Fig. 4(d)) does not largely affect superconducting properties of REBa2Cu3O7-d, which is a twodimensional layered compound. Because the trend is clearly different to that observed for cubic (NaCltype) tellurides with a HEA-type site, crystal-structure dimensionality is a key factor to how the introduction of HEA-type site affects superconducting properties in compounds. In Fig. 8(b), we found three data points for HEA-type samples show a Jc larger than that for the other low-entropy samples at OP = 0.01-0.015. Although it has not been fully clarified whether the slightly large Jcs in the HEA-type samples are caused by high-entropy alloying or not, the effect of high-entropy alloying for cuprates should be further studied to find the way to improve practical performance of cuprate superconductors.

BiS2-based layered superconductors RE(O,F)BiS2
BiS2-based superconductor family is one of the layered superconductor families and was discover in 2012 [49][50][51]. The crystal structure is composed of alternate stacks of a conducting BiS2 bilayer and a blocking layer (for example, a REO layer), which is similar to that of high-Tc systems.
Furthermore, unconventional superconductivity has been proposed from theoretical and experimental studies on the BiS2-based compounds [51].
A typical BiS2-based system is REOBiS2 (see Fig. 4(e)). Because non-doped REOBiS2 is a semiconductor, electron carrier doping is required to induce metallicity [50]. For RE = La, a superconducting transition was observed at 2.5 K after electron doping through partial substitution of O by F in LaO0.5F0.5BiS2. However, the superconductivity states in La(O,F)BiS2 is not bulk in nature.
This is due to the presence of the local disorder due to Bi lone pairs, and the local disorder could be suppressed by in-plane chemical pressure effects [52][53][54]. In-plane chemical pressure can be generated by RE-site substitution by smaller RE ions or Se substitution for the S site. By increasing in-plane chemical pressure, local disorder is suppressed, and bulk superconductivity is induced [52,53].
Therefore, one can say that RE(O,F)BiS2 is a useful system to investigate the effect of structural modification on local crystal structure and superconducting properties. This suggests that the investigation of the effects of introduction of a HEA site in RE(O,F)BiS2 would provide us with key information about interlayer interaction through the HEA states. composition. As shown in Fig. 9(c), these four samples have different lattice constants a, which is corresponding to different in-plane chemical pressures. Therefore, the variation of Tc can be understood with the in-plane chemical pressure scenario. When focusing on the samples with the same lattice constant a and different mixing entropy, we find that the superconducting properties (Tc and Δ4πχ) of the HEA-type sample are higher than those of low-entropy samples (Figs. 9(c) and 9(d)).
The facts indicate that an increase in ΔSmix for the RE site may positively work in improving superconducting properties. Therefore, we systematically prepared REO0.5F0.5BiS2 samples with the same lattice constant a and different ΔSmix to investigate the interlayer interaction [38]. As summarized in Fig. 10(a), we succeeded in preparation of a set of five REO0.5F0.5BiS2 samples with almost the same a and systematically varied ΔSmix. Because the lattice constant a nearly corresponds to the degree of the in-plane chemical pressure, the set of samples have almost similar inplane chemical pressure. Therefore, we could detect the effects of the increase in ΔSmix to