Recent advances in high-entropy superconductors
Introduction
Initial investigations into multielement materials, particularly high-entropy alloys (HEAs), were reported in 20041. The principle was subsequently extended to high-entropy ceramics (HECs), with the initial examples in powder form being high-entropy oxides published in 20152,3. Since then, numerous other materials have been characterized, leading to the adoption of the term high-entropy materials (HEMs) to more comprehensively describe this novel class of inorganic solids3,4. In general, Gibbs free energy is conventionally utilized to assess the phase stability of HEMs, with ∆Gmix = ∆Hmix – T∆Smix, ∆Gmix, ∆Hmix, and ∆Smix are the differences among the Gibbs free energy, mixing enthalpy, and mixing entropy, respectively. The ∆Smix of high-entropy systems are composed of four types of entropy. Multiple studies have indicated that ∆Smix is determined by the configurational entropy ∆Sconf. For a random n-component solid solution in which the ith component has a mole fraction of ci, its ideal ∆Sconf per mole is the following: (Delta {S}_{{conf}}=,-Rmathop{sum }nolimits_{i=1}^{n}{c}_{i}mathrm{ln}{c}_{i}), where n represents the number of components and R is the gas state constant. When ∆Sconf exceeds 1.61 R, the system is classified as high-entropy; when it falls within the range of 0.69 R and 1.61 R, the system is categorized as medium-entropy.
HEMs have garnered remarkable attention across various fields because of their excellent chemical, physical, and mechanical properties3,4. These materials present opportunities for customizing electronic properties, including superconductivity. Typically, weak disorder in nonmagnetic superconductors does not result in a substantial change in the critical temperature (Tc) for superconductivity5. The first HEA in which this phenomenon was observed was Ta34Nb33Hf8Zr14Ti11, reported in 20146. Since then, numerous studies on high-entropy superconductors have been conducted, driven in part by their combined mechanical and electronic properties in various applications7,8. However, modifying the composition of HEMs does not consistently result in the same or enhanced properties. Therefore, the advantages and challenges associated with a high-entropy structure need to be meticulously evaluated.
The susceptibility of superconductivity to interruption by external fields and local defects in practical applications must be considered. Consequently, the development of superconductors that are robust against environmental perturbations has been a long-standing objective. In addition, the Majorana fermions in topological superconductors have been proposed as a potential foundational qubit for quantum computation, which shows strong resistance to local noise. The essential features of a high configurational entropy are lattice distortions, cocktail effects, improved stability of the crystal structure, and high defect density3,4. Therefore, the use of high-entropy superconductors in extreme environments is promising.
Accordingly, the present review is dedicated to summarizing the recently reported works on the structural type and physical properties of high-entropy superconductors, as well as their potential applications. We provide our perspective on the future challenges of high-entropy superconductors.
Types of high-entropy superconductors and research progress
BCC-type high-entropy superconductors
Figure 1 shows representative high-entropy crystal structures with the atoms randomly arranged themselves in the crystallographic positions. The first HEA superconductor, Ta34Nb33Hf8Zr14Ti11, possesses the unit cell parameter a = 3.36 Å in the body centered cubic (BCC) structure6. Experimental measurements indicated that Ta34Nb33Hf8Zr14Ti11 is a type II superconductor characterized by a Tc = 7.3 K, a lower critical field μ0Hc1(0) = 32 mT, an upper critical field μ0Hc2(0) = 8.2 T, and an energy gap in the electronic density of state (DOS) at the Fermi level of 2∆ = 2.2 meV. Analysis of various criteria using parameters of superconductivity suggests that Ta34Nb33Hf8Zr14Ti11 is close to a phonon-mediated, BCS-type superconductor within the weak electron‒phonon coupling limit. After superconductivity was observed in Ta34Nb33Hf8Zr14Ti11, extensive and systematic studies were conducted. The Ta-Hf-Nb-Zr-Ti system has emerged as the most extensively investigated BCC-type HEA superconductor9,10,11,12,13,14,15,16,17. In BCC-type HEA superconductors, the Tc range is approximately 2–9 K, and the valence electron count (VEC) range is approximately 3.2–5.26,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28. Figure 2 shows representative superconducting behavior in one of the BCC-type HEAs, as demonstrated by transport measurements23.
a–f BCC, CsCl, HCP, α-Mn, A15, and layer structures with randomly distributed atoms.
a Magnetization curves of TiHfNbTaMo. b M(H) data in the temperature range of 1.8–3.5 K under applied fields of 0–500 Oe. c The temperature dependence of the lower critical field. d Electrical resistivity at 0 T. The inset shows the field dependence of the resistivity between 0 and 3 T around the Tc. e The temperature-dependent μ0Hc2 was fitted with the GL model. f Temperature dependence of the electronic-specific heat. Reproduced with permission23. Copyright 2023, Wiley-VCH.
The elemental composition is an essential consideration in the design of BCC-type HEA superconductors. The trend observed in crystalline transition metal superconductors is described by the Mattias rule, which indicates that the Tc achieves its maximum value at a VEC of approximately 4.6729. Studies have revealed that the VEC of BCC-type HEA superconductors significantly influences both their superconducting Tc and material stability7,23. The BCC-type HEA superconductors predominantly consist of transition metals such as Nb, Ta, Zr, V, Hf, and Ti, which exhibit VECs of either 4 or 57,8. The investigation of BCC-type HEA superconductors with VEC values exceeding 4.8 is of considerable interest, given that this specific range remains relatively underexplored. The systematic investigation of BCC-type HEA superconductors with VEC values greater than 4.8 is still emergent. BCC-type HEA superconductors with VECs of 4.8 or more have been reported for (TiZrNbTa)1-xWx30, Nb20Re20Zr20Hf20Ti2019, NbTaTiZrFe31, TiHfNbTaMo23, and Ti-Hf-Nb-Ta-Re systems25. Moreover, BCC-type HEA superconductors possessing VECs exceeding 4.8 offer an additional material platform for investigating the correlation between superconducting properties and VEC.
Superconductivity has been observed not only in BCC-type HEAs composed of d-electron elements but also in alloys incorporating uranium or thorium, namely, (TaNb)0.31(TiHfU)0.69 and (TaNb)0.67(MoWTh)0.3332,33. This discovery broadens the scope of the search for new BCC-type HEA superconductors for alloys with actinides and possibly lanthanides. The impact of hydrogen on the superconducting characteristics of BCC-type HEAs and MEAs has also been reported. The measurements show that hydrogenation leads to the suppression of superconductivity in BCC-type HEAs and MEAs34. In addition, different heat treatment conditions, such as annealing, quenching, and combination processes, affect the superconducting properties and microstructures of BCC-type HEAs27,35. Annealing conditions affect the critical current density Jc more significantly than the Tc and upper critical fields do, with a significant improvement in the flux pinning force density Fp. The Jc at 4.2 K of the annealed Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 HEA at 550 °C for 24 h is improved significantly by approximately 1860% compared with that of the as-cast sample36. In addition, the fishtail or second peak effect was determined for the BCC-type (NbTa)0.7(TiZrHf)0.5 superconductor. Notably, Fp reaches a maximum at a reduced field value of 0.72, which markedly contrasts with the behavior observed in iron-based and cuprate high-Tc superconductors17.
FCC-type high-entropy superconductors
The face centered cubic (FCC) structure of HEA remains stable as the VEC exceeds 8, in accordance with a hard sphere model. Below that value, a change in the stacking from FCC-type stacking to hexagonal close-packed (HCP) type stacking is expected. However, according to Matthias’ rule, superconductivity rarely occurs in stable FCC-type HEAs with a VEC higher than 8 and a BCC phase with a VEC smaller than 6.87. The intermediate VEC range of 6.87–8 corresponds to mixed phases of FCC and BCC37,38,39. However, by adding carbon with a VEC of 4 to the HEA, a single FCC-type HEA phase can be stably formed in a VEC between 6.2 and 7. Superconductivity was observed in the (MoReRu)(1-2x)/3(PdPt)xCy system, which is the first FCC-type HEA superconductor40. FCC-type HEA superconductors were also reported in the M-Pt-Ir-Rh-W (M = Re, Ta, Nb, and Mo) single-phase system, which was synthesized via splat cooling with a VEC between 7.87 and 8.3841. The superconducting properties of this system follow the general trends published for metallic alloys. However, the highest Tc is only approximately 1.5 K in the M-Pt-Ir-Rh-W system. HEA superconductors with FCC structures is a research topic in this field.
NaCl-type high-entropy superconductors
The concept of high entropy is also used in tellurides to explore new high-entropy superconductors. Several metal tellurides exhibit NaCl-type structures. Since the structure of a NaCl-type telluride comprises both an anionic tellurium site and a cationic metal site, high-entropy telluride presents an ideal material for investigating the influence of high entropy on the states of a tellurium site. A series of polycrystalline samples of the high-entropy telluride AgInSnPbBiTe5 and (Ag, In, Pb, and Bi)Te1-xSex were synthesized via high-pressure synthesis42,43. The superconducting transition was observed at approximately 2.6 K for AgInSnPbBiTe5. This finding is useful for the further development of various high-entropy compounds with superconducting properties. Recently, a series of superconductors have been discovered in NaCl-type structure high-entropy ceramics44,45,46. Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C, the first high-entropy carbide superconductor, shows bulk type II superconductivity with Tc = 2.35 K, μ0Hc1(0) = 26.1 mT, and μ0Hc2(0) = 0.51 T44. Superconductivity was also observed in the Ti0.2Zr0.2Nb0.2Ta0.2Mo0.2Cx and Ti0.2Nb0.2Ta0.2Mo0.2W0.2C1-xNx systems45,46. The highest Tc of these high-entropy carbide superconductors is approximately 6 K, which is observed in the Ti0.2Nb0.2Ta0.2Mo0.2W0.2C0.7N0.3 sample. The discovered high-entropy carbide superconductors are type II superconductors, with μ0Hc2(0) values between 0.5 and 5 T. The specific heat jump value is close to the theoretical BCS value of 1.43.
CsCl-type high-entropy superconductors
A prevalent cubic structural type observed in simple intermetallic AB compounds is the CsCl-type structure, which is analogous to the BCC structure. However, unlike the random distribution of atoms at the corners and center of the cell, the A and B atoms are ordered separately, occupying distinct positions and forming interpenetrating primitive cubic lattices. Superconductivity in CsCl-type HEAs was first reported in hexanal (TaNbZrSc)1-x(PdRh)x and (NbZrSc)1-x(PdRh)x samples47. The (NbZrSc)0.65(PdRh)0.35 HEA has a maximum Tc of approximately 9.3 K and has the largest μ0Hc2(0) of 10.7 T, which is similar to that of the NbTi superconductor47. The VEC dependence of Tc in these systems does not conform to the behavior observed in either crystalline or amorphous transition metal superconductors. This deviation is attributed to the structural complexity of CsCl-type HEAs, which exhibit a distinct trend. Additionally, the computed electronic lifetime τ is very short in (NbZrSc)1-x(PdRh)x samples, resulting in a small mean free path of electrons on the order of interatomic distance, which positions (NbZrSc)1-x(PdRh)x near the Mott–Ioffe–Regel limit48.
HCP-type high-entropy superconductors
The majority of HCP-type HEAs reported in the literature contain rare-earth atoms. In contrast to transition metal HEAs, these alloys lack solid solution strengthening effects, which is attributable to mismatches in the atomic size and electronic structure among the component elements49,50. Compared with BCC-type HEA superconductors, HCP-type superconductors are less reported. The first HCP-type superconducting HEA, Re0.56Ti0.11Zr0.11Hf0.11Nb0.11, displays bulk superconductivity with Tc = 4.4 K51. Measuring the specific heat at low temperatures reveals that Re0.56Ti0.11Zr0.11Hf0.11Nb0.11 functions as a phonon-mediated superconductor in the weak electron‒phonon coupling limit, characterized by a normalized specific heat jump of ∆Cel/γnTc = 1.32. Bulk superconductivity was also observed in HCP-type Nb10+2xMo35-xRu35-xRh10Pd10, Mo35WxReyRuzPd5, and (MoReRu)(1−2x)∕3(PdPt)x40,52,53. A Tc of approximately 8.32 K was observed in the Mo35W10Re20Ru30Pd5 HCP-type superconductor, which currently has the highest Tc among HCP-type HEA superconductors53. In addition, studies have shown that an optimal VEC for an HCP-type MEA/HEA is close to 7.0 and that the Tc is unlikely to exceed 10 K54. Future research can determine whether increasing the Tc of HCP-type HEA superconductors is possible through the discovery of the optimal VEC and various elemental constituents.
α-Mn-type high-entropy superconductors
The α-Mn structure originates from the low-temperature allotrope of manganese. Compared with simple BCC and CsCl-type structures, the α-Mn structure has a body-centered cubic crystallographic cell that contains a relatively large number of atoms. Superconductivity was observed in the (HfTaWPt)1-xRex, (IrWTaHf)1-xRex, and (ZrNb)1-x(MoReRu)x α-Mn-type systems55. All of these systems show type II superconductivity with strongly varying Tcs depending on the VEC and cubic lattice parameter a. The Tcs increase linearly with increasing VEC and decreasing a within each system and fall between the trend lines observed in amorphous and crystalline transition metal alloy superconductors. The VEC of HEA superconductors with α‒Mn structures ranges from 6.5 to 6.956.
A15-type high-entropy superconductors
V1.4Nb1.4Mo0.2Al0.5Ga0.5 is an A15-type HEA superconductor with Tc = 10.2 K and a disorder-enhanced μ0Hc2(0) of approximately 20 T, which are the highest recorded values among HEA superconductors under ambient conditions57. The specific heat data show that the Debye temperature and specific heat jump values of Nb3Sn0.3Al0.3Ge0.2Ga0.1Si0.1 and Nb3Sn0.2Al0.2Ge0.2Ga0.2Si0.2 of the A15-type structure are close to those of traditional Nb3Sn and V3Si superconductors58,59.
Other types of high-entropy superconductors
High-entropy antimonide M1-xPtxSb (M = equimolar Ir, Pd, Rh, and Ru) compounds crystallized in pseudohexagonal NiAs-type structures with the space group P63/mmc have been prepared via a conventional solid-state reaction followed by quenching60. Bulk superconductivity at Tc = 2.15 K was discovered for the compound with x = 0.2. This Tc is as high as the highest Tc previously reported for transition-metal monoantimonides, suggesting that the high-entropy principle offers a new method for discovering new superconducting materials and improving the Tc.
High-entropy superconductivity is also found in layered structures61,62,63,64, σ-type structures65,66, and CuAl2-type structures67. In addition, high-entropy superconductors have already been introduced into high-temperature superconductors, and a series of high-entropy oxides, REBa2Cu3O7-δ (RE is lanthanide), with a Tc of approximately 92 K, were successfully synthesized68.
Factors that influence the T
c
Valence electron count
Experimental evidence has indicated that the VEC of HEA and HEC superconductors is critical in determining their superconducting Tc and material stability. Despite the significant disorder and randomness in HEMs, their superconducting properties are markedly influenced by their chemical composition and complexity. Figure 3 displays the Tc values of (NbTa)1-x(TiHfZr)x, Ti0.2Nb0.2Ta0.2Mo0.2W0.2C1-xNx, and Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C as a function of VEC9,44,46. The Tc range is approximately 2–8 K in BCC-type HEA superconductors. The highest Tc is reached when the VEC is approximately 4.67, which is the essential feature of the Matthias rule29. The Tc values of (NbTa)1-x(TiHfZr)x superconductors fall between those of crystalline alloys and amorphous materials. In NaCl-type high-entropy carbide superconductors, the VEC is 4.2 for Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C, and Ti0.2Nb0.2Ta0.2Mo0.2W0.2C1-xNx (0 ≤ x ≤ 0.45) has a VEC ranging from 4.6 to 4.825. The highest Tc of 6.03 K was observed for Ti0.2Nb0.2Ta0.2Mo0.2W0.2C0.7N0.3, with VEC = 4.75, which is also in agreement with the Matthias rule. The observed increase in Tc in NaCl-type high-entropy carbide superconductors is relatively less pronounced than that in crystalline alloys, exhibiting a monotonically increasing trend akin to that observed in amorphous superconductors. Moreover, the relationship between VEC and Tc in NaCl-type high-entropy carbides is highly consistent with that in BCC-type (TaNb)1-x(ZrHfTi)x HEA superconductors.
VEC dependency of the Tc for amorphous 4 d metal, crystalline 4 d metal, (NbTa)1-x(TiHfZr)x, and high-entropy carbide systems9,29,44,46.
Mixing entropy and elemental composition
The high mixing entropy caused by the atoms arranged randomly on the lattice is a unique feature of HEMs. Experimental measurements have shown that several BCC-type alloy superconductors have different Tc values with the same VEC = 4.79,69. There is no loss of superconductivity or substantial reduction in Tc with the disorder introduced resulting from an increased number of constituent elements. Thus, the atomic disorder present in the BCC-type HEAs has a minimal influence on the superconducting state, as Tc is insensitive to disorder in both simple crystalline and amorphous metallic states. Furthermore, the upper critical fields appear to increase with increasing entropy in some high-entropy systems9,44,70.
Although the VEC of most BCC-type HEAs with a Tc > 7 K is approximately 4.67, the Mattias rule cannot describe the correlation between Tc and VEC for different BCC-type HEA systems, which possess different element compositions. The elemental composition seems to have a strong influence on the Tc for BCC-type HEA superconductors. The superconductivity of (NbTa)0.67(TiZrHf)0.33 is replaced by electron mixtures such as [Sc-Mo], [Y-Mo], and [Sc-Cr], and the Tc in (NbTa)0.67(TiZrHf)0.33 significantly decreases after Nb or Ta are replaced by isoelectronics, while the effects of Hf, Zr, and Ti substitution on Tc are limited71. In BCC-type HEA superconductors, the content of Nb has a significant influence on the Tc, with a higher Nb content possibly resulting in a higher Tc23. The presence of Nb significantly increases the Tc with the same VEC in BCC-type HEA superconductors. The Nb content in BCC-type HEAs is typically 25% or higher when the Tc is > 7 K6,9,12,13,23,71. This observation appears predictable because Nb has the highest Tc of 9.2 K among all the elements at ambient pressure72. Hence, a VEC of approximately 4.67, coupled with a substantial proportion of Nb, is conducive to increasing the Tc for BCC-type HEAs.
Pressure
As a conventional thermodynamic parameter, pressure is an effective and suitable means of tuning the electronic properties of solids. By triggering structural and electronic transitions, novel quantum phenomena can be induced73,74,75. In the context of quantum materials, the pressure-induced enhancement of Tc in iron-pnictide and copper oxide superconductors76,77,78 and the pressure-induced superconductivity in H3S79,80, nickelate La3Ni2O781, and other elements82,83 are examples. Therefore, understanding what happens to pressurized high-entropy materials is of great interest. Remarkably robust zero-resistivity superconductivity was observed in the pressurized BCC-type (TaNb)0.67(HfZrTi)0.33 HEA (as shown in Fig. 4)10. The Tc increases from 7.7 K at ambient pressure to 10 K at approximately 60 GPa and then subsequently decreases to 9 K at 190.6 GPa, a pressure comparable to that found in the outer core of the Earth. Moreover, the μ0Hc2(0) of (TaNb)0.67(HfZrTi)0.33 is approximately 8 T at ambient pressure, approximately 4 T at 100 GPa, and approximately 2 T at 179.2 GPa. The BCC-type structure is still maintained up to pressures of approximately 96 GPa. A high ∆Smix decreases the Gibbs free energy, thereby improving phase stability. These unique physical phenomena make the (NbTa)0.67(TiZrHf)0.33 HEA a promising superconductor for new applications under extreme conditions. The robustness of the superconducting state was also observed in AgInSnPbBiTe5 high-entropy telluride under high pressure84. The Tc of AgInSnPbBiTe5 with a CsCl-type structure is almost independent of pressure for pressures ranging from 13 to 35 GPa.
Phase diagram of (TaNb)0.67(HfZrTi)0.3340, Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C45, AgInSnPbBiTe584, and TaNbHfZr85 as a function of pressure and Tc.
As a new class of high-entropy superconducting materials, the regulation of superconductivity by pressure in high-entropy carbides has also been investigated. The Tc slightly decreases from 2.67 K at ambient pressure to 2.15 K at approximately 50 GPa in the Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C superconductor44. The superconductivity is robust, with a Tc variation of merely 0.5 K across a pressure range of 50 GPa, akin to that observed in the (NbTa)0.67(TiZrHf)0.33 HEA. The T-dependent resistance of Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C shows interesting scaling behavior. With increasing pressure, the s band shifts toward higher energy levels, facilitating s-d electron transfer, increasing the number of d electrons, and strengthening the electron correlation effects. In addition, the Tc remains almost constant under different pressures (0–80 GPa), which is also observed for the Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C superconductor45.
The pursuit of high Tc superconductivity in transition metal alloys is particularly challenging, as d electrons are commonly not favored by conventional BCS theory82. The Tc of most HEA and MEA superconductors does not exceed 10 K. Recently, a new record Tc was observed in a BCC-type TaNbHfZr MEA superconductor under pressure85. Under pressure, Tc exhibits a dome-shaped curve in the phase diagram, which is different from that of (NbTa)0.67(TiZrHf)0.33 HEA and high-entropy carbides. The Tc is 8.1 K at ambient pressure and achieves a broad maximum of approximately 15 K at 70 GPa before decreasing to 9.3 K at approximately 157 GPa. Two competing effects are attributed to the dome-shaped curve in DFT calculations: the simultaneous suppression of the electron‒phonon coupling constant λ and the enhancement of the logarithmically averaged characteristic phonon frequency ωlog. Furthermore, the phonon dispersion shows no imaginary frequencies up to 155 GPa, indicating the stability of the BCC-type phase in the TaNbHfZr superconductor under high pressure. Therefore, the pressure-dependent Tc may display different features from those obtained through changing the components, which will help us understand the intrinsic behavior of superconductivity.
MPX3 (X = S, Se, M = divalent metal) is a large family of magnetic van der Waals materials that has various performances in applied and fundamental research86. The medium-entropy MPX3 (M = In, Cd, Mn, and Fe) was prepared via an entropy enhancement strategy87. The P‒P dimers recombine under external pressure, as demonstrated through the dramatic collapse of the c axis in Fe0.8Mn0.1Cd0.05In0.03PSe3 and (FeMnCd)0.25In0.17PSe3, which induces superconductivity simultaneously. The maximal Tc is 6.2 K at approximately 30 GPa for the Fe0.8Mn0.1Cd0.05In0.03PSe3 compound. The medium-entropy MPX3 (M = In, Cd, Mn, and Fe) shows a rich electron phase diagram, which transitions from a spin-glass insulator to a metal and then to a superconductor.
Both the robustness of the Tc to high pressure and the superconducting phase diagram that forms dome-shaped, high-entropy superconductors are highly resistant to physical high pressure. High-entropy superconductors, therefore, are promising for applications in extreme conditions and pose challenges for verifying the known superconductivity theory and developing a new theory.
Unconventional superconductivity in high-entropy superconductors
Although most HEAs are conventional phonon-mediated s-wave superconductors, some HEAs exhibit unconventional superconducting properties. The superconducting gaps of the BCC-type Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 and Nb2/5Hf1/5Zr1/5Ti1/5 compounds, as determined by the semiempirical α-model from specific heat measurements, exceeded the BCS weak coupling limit, suggesting that they are strongly coupled superconductors12,13. Figure 5 shows the heat capacity data for the Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 superconductor. Despite the superconducting properties of the Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 and Nb2/5Hf1/5Zr1/5Ti1/5 compounds falling within the dirty limit, the relatively large ∆Cel/γnTc values indicate that strongly correlated behavior exists beyond the BCS theory. The Kadowaki–Woods ratios (A/γ2) of the Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 and Nb2/5Hf1/5Zr1/5Ti1/5 compounds closely approximate those observed in heavy fermion systems characterized by a significant effective mass. The conflicting characteristics observed between conventional BCS-type metallic superconductivity and strong correlations may originate from the dirty superconducting regime with high Fermi velocity. In addition, extremely strong coupling behavior was also observed in TiHfNbTa superconductors16. The normalized Cel jumps at ({T}_{c}^{{mid}}) (∆Cel/γn({T}_{c}^{{mid}})) and ({T}_{c}^{{zero}}) (∆Cel/γn({T}_{c}^{{zero}})) for the TiHfNbTa superconductor are 2.88 and 2.13, respectively. Both ∆Cel/γnTc values in the TiHfNbTa superconductor significantly exceed the BCS weak coupling ratio (1.43), thereby suggesting strong coupling in TiHfNbTa. The strong coupling behavior and strongly correlated electron behavior in s-wave superconductors are noteworthy because many superconductors in strongly correlated systems are heavy fermions and high-Tc cuprate superconductors, which exhibit nodal superconductivity.
a C/T vs. T2 curves for the Nb2/6Ta1/6Hf1/6Zr1/6Ti1/6 superconductor. b Specific heat data over a temperature range of 1.8–9 K. Reproduced with permission12. Copyright 2020, Elsevier.
A further signature of the unconventional nature of HEA compounds is that μ0Hc2(0) is greater than the Pauli paramagnetic limit in a series of noncentrosymmetric cubic β-Mn-type Cr5+xMo35-xW12Re35Ru13C20 (0 ≤ x ≤ 9) HEA superconductors, and the μ0Hc2(0) approaches or even slightly exceeds its Pauli paramagnetic limit88. These β‒Mn-type HEAs show bulk superconductivity, with Tc decreasing monotonically from 5.49 K to 3.35 K as the concentration of x increases. This reduction in Tc is attributed to pair-breaking effects caused by the local moments of Cr. Spin‒triple pairing might contribute to the large upper critical field for noncentrosymmetric superconductors. However, this scenario is improbable given that spin‒triplet pairing is disfavored in highly disordered HEA systems88. The μ0Hc2(0) exceeding the Pauli paramagnetic limit was also found in BCC-type ScVTiHfNb21, HCP-type Ru0.35Os0.35Mo0.12W0.1Zr0.189, A15-type (Nb0.5V0.5)3-xMoxAl0.5Ga0.557, and α‒Mn-type Ru0.35Os0.35Mo0.1W0.1Zr0.1 HEA superconductors89. An applied magnetic field greater than μ0Hc2(0) can decrease the superconductivity. Type II superconductors can be carried out via two mechanisms: the Pauli paramagnetic effect and the orbital limiting effect. The Maki parameter αM is 1.05 for the ScVTiHfNb HEA superconductor21. Depending on the αM value, pair breaking is due primarily to the Pauli paramagnetic effect, with a small contribution from the orbital limiting effect. The metallic nature coupled with a high μ0Hc2(0) makes them attractive candidates for application in superconducting devices.
In addition, the majority of research has concentrated on the synthesis of HEA superconductors using combinations of transition metal elements. However, the potential incorporation of actinide or lanthanide elements into these HEAs is worth exploring. The f-electron elements, in particular, show valence flexibility, a propensity to hybridize with conduction electrons, and significant spin anisotropy. Therefore, HEAs that introduce f-electrons may exhibit unconventional superconducting properties. The (NbTa)0.31(TiHfU)0.69 is the first and only HEA superconductor that contains an actinide element32. Like (NbTa)0.31(TiHfZr)0.69, this HEA crystallized in a BCC-type lattice and exhibited phonon-mediated superconductivity with a Tc = 3.2 K. However, the μ0Hc2(0) of (NbTa)0.31(TiHfU)0.69 is approximately 6.4 T, which exceeds the Pauli paramagnetic limit of 5.95 T. This observation implies that strong spin‒orbit coupling may influence the superconductivity.
In some HEA systems, high-entropy superconductors appear to have higher μ0Hc2(0)/Tc values than binary or ternary compounds44,70. The μ0Hc2(0)/Tc is shown to be proportional to γρN in the dirty limit. It is plausible that the increases in ρN, attributable to increased disorder, are primarily responsible for the increase in μ0Hc2(0)/Tc. However, quantifying the intrinsic ρN values remains challenging owing to the polycrystalline nature of high-entropy samples, where resistivity is largely influenced by grain boundary contributions57,88. The composition of HEAs, which are devoid of high atomic concentrations of specific elements, is a promising avenue for developing superconductors with elevated μ0Hc2(0). The HEA concept can facilitate the development of new superconducting materials, including HEA sites and/or HEA-type layers. The high-entropy layer single crystals, ROBiS2 (R = Sm + Nd + Pr + Ce + La), have been successfully grown via CsCl flux64. In this high-entropy layer system, the μ0Hc2(0) in the c-plane is significantly greater than its Pauli limit. HEM superconductors, known for their disorder, provide a unique opportunity for exploring the superconducting pairing mechanism more clearly.
Topological bands
The combination of superconductivity and topology is expected to display novel types of quasiparticles, including non-Abelian Majorana zero modes or fractional charge and spin currents90,91,92. The experimental realization of topological superconductivity will offer excellent platforms for developing fault-tolerant quantum computing techniques. Currently, topological properties are reported in many high-entropy carbide superconductors44,45,46. Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C is the first reported superconductor in a high-entropy carbide with Tc = 2.35 K. Figure 6a–f shows orbital-resolved band structures of six representative special quasirandom structures (SQSs) of Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C, incorporating the effects of spin‒orbital coupling (SOC). The DFT calculations show that Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C has six type II Dirac points, which are attributed mainly to the p orbitals of carbon and the t2g orbitals of transition metals44. These characteristics suggest that the type II Dirac points in Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C are closely associated with the band inversion at the Γ point, analogous to the superconductor TaC. When accounting for the impact of disorder from the transition metal elements on the band structures of Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C, the energies of the type-II Dirac points deviate from the Fermi level, ranging between −0.231 eV and −0.017 eV. (as shown in Fig. 6g–i). Notably, the energies of the type II Dirac point away from the Fermi level are −1.0 and −1.3 eV in NbC and TaC93,94, respectively, which are much lower than those of Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C.
a–f Orbital-resolved band structures of six representative SQSs when the SOC is included in the DFT calculations. The type-II Dirac points (DPs) are indicated by green arrows. The Fermi level is set to zero in (a–f). The energies of the type-II DPs when they are located along X-Γ (g), Γ-Y (h), and Γ-Z (i). j The dependence of the energies for the type II DPs on the compressive strain in Ti0.2Zr0.2Nb0.2Hf0.2Ta0.2C. Reproduced with permission44. Copyright 2023, Wiley-VCH.
Furthermore, the type II Dirac points are precisely located at the Fermi level when compressive pressure is applied. Nontrivial electronic bands also exist in other high-entropy carbide superconductors, such as Ti0.2Zr0.2Nb0.2Ta0.2Mo0.2C and Ti0.2Nb0.2Ta0.2Mo0.2W0.2C1-xNx (0 ≤ x ≤ 0.45)45,46. In the Ti0.2Nb0.2Ta0.2Mo0.2W0.2C1-xNx systems, the Dirac points will be farther from the Fermi level with increasing N content. High-entropy carbides offer a novel platform for studies of the coupling between superconductivity and topological physics. These findings imply that high-entropy materials offer a viable pathway for engineering band structures and broadening the scope of topological materials.
High-entropy film superconductors
In 2020, BCC-type superconducting (NbTa)1-x(TiHfZr)x HEA films were fabricated through magnetron sputtering at room temperature, achieving an average thickness of 600–950 nm95. For films with x ≥ 0.65, the normal state results in weak insulating properties, whereas for x < 0.65, the films exhibit metallic behavior. All (NbTa)1-x(TiHfZr)x HEA films, with the exception of those with x = 0 or 1, exhibit a superconducting phase transition at low temperatures. The highest value of Tc = 6.9 K was obtained for x = 0.43. The highest μ0Hc2(0) is 11.05 T, occurring at an approximately equimolar ratio of x ~ 0.65, with the largest mixing entropy. The superconducting properties obtained from the experimental measurements of all the films are close to those reported for amorphous superconductors, indicating the potential for the development of superconducting nanowire single-photon detectors. Interestingly, μ0Hc2(0) surpasses the Pauli paramagnetic limit by up to 20% for x > 0.6. This exotic high μ0Hc2(0) in those HEA films may be attributed to strong electron‒phonon coupling, as the likelihood of enhanced spin‒orbital coupling effects appears to be unlikely.
In addition, the crystalline and amorphous phases affect the physical properties of HEA films. A crystal/glass phase is formed in the near equimolar region of (NbTa)1-x(TiHfZr)xMoy and (NbTa)1-x(TiHfZr)x HEA films deposited through magnetron sputtering96, which involves the coexistence of nanocrystalline granules and amorphous aggregations. The onset of superconductivity in crystal/glass HEA films is dominated by the nanocrystalline phase, whereas the amorphous aggregation phase limits the zero-resistivity superconducting state, resulting in a two-step normal-to-superconducting phase transition97. Moreover, the Meissner effect in these HEA films is determined exclusively at temperatures significantly below the onset of superconductivity. Unusual superconducting transitions are detected only in the films with high mixing entropies, indicating that the specific compositional range substantially influences the collective electronic properties.
High-quality superconducting thin films of Nb–Ta–Ti–Zr–Hf HEAs have also been prepared via pulsed laser deposition98. The Nb-Ta-Ti-Zr-Hf HEA thin films exhibited a large Jc of > 1MA cm-2, which was approximately 790 and 820% greater than that of the bulk sample at 4.2 K and 2.0 K, respectively (as shown in Fig. 7). Additionally, these films demonstrate exceptionally robust superconductivity to irradiation-induced disorder modulated by the dose of Kr-ion irradiation. Notably, these HEA films exhibit resistance to displacement damage that is more than 1000 times greater than that of other promising superconductors with technological applications, such as Nb3Sn, MgB2, iron-based superconductors, and high-Tc cuprate superconductors99,100,101,102. Recent studies have determined that high entropy can significantly improve the in-field properties of high-entropy cuprate films. The maximum Fp of high-entropy cuprate films at 77 K is 1.7 times greater than that of undoped YBCO films103. Therefore, high-entropy thin films are consequently favorable for superconductivity devices as well as large-scale applications, such as high-field superconductivity magnets, aerospace applications, and irradiation environments.
a, b Magnetic field dependence of Jc for HEA thin films at 2.0 and 4.2 K, respectively. The red dashed line indicates Jc = 100 kA cm−2, a standard benchmark for large-scale applications, including high-field superconducting magnets. c The comparative Jc values at 2.0 K and 4.2 K under a magnetic field of 1 T demonstrate the potential of these HEA films for extensive practical applications.
A nitrogen-doped strategy is commonly used to increase the superconductivity of materials. The influence of nitrogen incorporation on the physical properties of TiNbMoTaWNx and TiZrTaNbNx HEA films has been reported104. The TiNbMoTaWNx HEA films have a dome-shaped Tc on x, with the highest Tc reaching approximately 5 K at an optimal x value of 0.74, where Tc is 0.6 K for x = 0 and 1 K for x = 0.97. High μ0Hc2(0) values that surpass the Pauli paramagnetic field are also observed in this system. Further investigations via point-contact spectroscopy and heat capacity measurements confirmed that the superconductivity in these HEA nitride films is bulky in nature. The incorporation of nitrogen into HEAs presents a promising strategy for tuning their superconducting properties, particularly the Tc105. In addition to chemical doping, the Tc of HEA films is also affected by the deposition temperature in the DC magnetron sputtering method30. Increasing the deposition temperature to 400 °C can increase the Tc of the (TaNbZrTi)1-xVx and (TaNbZrTi)1-xWx HEA films.
Summary and perspective
The superconducting properties of HEMs are markedly different from those of conventional alloy superconductors, copper oxide superconductors, iron-based superconductors, and amorphous superconductors, indicating that HEMs can be regarded as a separate class of superconducting materials. To systematically review previous research and identify the primary areas of focus within this domain, we reviewed significant advancements in the field of high-entropy superconductors over the past decade. Some interesting phenomena, such as the robustness of superconductivity to pressure, a large upper critical field, strong coupling behavior, and a topological band structure, have been reported in high-entropy superconductors. The variability in superconducting behavior across different systems provides a unique opportunity to uncover additional physical phenomena through comparative analysis. Furthermore, to improve our understanding of the factors influencing the Tc of HEMs and to ascertain the general applicability of high-entropy superconductors, we have summarized various modulation methods across different material systems.
Despite the remarkable discoveries and rapid advancements in this field, many challenges still exist. This paper outlines our perspectives on the principal challenges and suggests several potential research directions in the research of high-entropy superconductors.
(1) HEA superconductors with higher Tc at ambient pressure is a topic of research in this field. HEA superconductors exhibit excellent mechanical properties, and HEA superconductors may be excellent candidates for fabricating superconducting magnets in the future.
(2) Unconventional superconducting properties (strongly correlated behavior) and topological properties have been detected in some high-entropy superconductors. To explain these behaviors, further investigation is needed. A strategy for further research on these high-entropy superconductors, including an examination of quantum critical behavior under external magnetic fields and gap-symmetry investigations, such as point contact spectroscopy, is recommended as a pathway toward the investigation of single-crystal growth and diverse physical properties in these HEMs.
(3) There are relatively few theoretical investigations into the phonon spectra and electronic structures of HEM superconductors. HEM superconductors also provide a good platform for theoretical scientists to analyze the superconductivity mechanism.
(4) More systems of high-entropy superconductors will be discovered and reported in the future, especially in the field of HECs (high-entropy carbides, high-entropy nitrides, high-entropy MAX phases, high-entropy MXenes, etc.). There are relatively few reports on HEC superconductivity, and similar to HEAs, HECs have much room for exploration. In addition, HECs may introduce configurational disorders into the anionic sublattice, thereby further increasing entropy and expanding the compositional space available for the discovery of new materials with unique properties.
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