Superconductivity in the High‐Entropy Ceramics Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C x with Possible Nontrivial Band Topology

Abstract Topological superconductors have drawn significant interest from the scientific community due to the accompanying Majorana fermions. Here, the discovery of electronic structure and superconductivity (SC) in high‐entropy ceramics Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C x (x = 1 and 0.8) combined with experiments and first‐principles calculations is reported. The Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C x high‐entropy ceramics show bulk type‐II SC with T c ≈ 4.00 K (x = 1) and 2.65 K (x = 0.8), respectively. The specific heat jump (∆C/γT c) is equal to 1.45 (x = 1) and 1.52 (x = 0.8), close to the expected value of 1.43 for the BCS superconductor in the weak coupling limit. The high‐pressure resistance measurements show a robust SC against high physical pressure in Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C, with a slight T c variation of 0.3 K within 82.5 GPa. Furthermore, the first‐principles calculations indicate that the Dirac‐like point exists in the electronic band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C, which is potentially a topological superconductor. The Dirac‐like point is mainly contributed by the d orbitals of transition metals M and the p orbitals of C. The high‐entropy ceramics provide an excellent platform for the fabrication of novel quantum devices, and the study may spark significant future physics investigations in this intriguing material.


Introduction
It is always keening for condensed matter scientists to discover new materials and explore their unique physical properties.[3][4] The experimental realization of topological SC will provide an excellent platform for developing faulttolerant quantum computing techniques. [5]wever, searching for topological superconductors (TSCs) has been challenging.
10][11][12][13][14][15][16] The existence of nontrivial topology in intrinsic superconducting materials offers the possibility to realize TSCs, preventing the complexity of fabricating a proximity-coupled heterostructure of a superconductor and topological insulator.[19][20][21][22] In addition, both predicted and experimental intrinsic TSCs are exceedingly rare.Most of them can only achieve SC or suitable topological surface states near the Fermi energy (EF) by doing.
It is highly urgent to search for more intrinsic TSC candidates with high superconducting critical temperature (Tc) and topological surface states near EF.2][33][34] Among these TSCs, TMCs have a relatively high Tc.The type-Ⅱ Dirac semimetal states were proposed to exist in the band structure of NbC and TaC, which are well-known comparable high Tc ~ 11.5 K and 10.6 K superconductors. [30,32,34]The first-principles calculations also indicate that s-wave Bardeen-Cooper-Schrieffer (BCS) SC with Tc ~ The high-entropy alloy (HEA) concept was developed in 2004 [35] , and since then, an entropy stabilization concept has been used to prepare high-entropy ceramics (HECs) as well. [36,37]HECs are the solid solution of five or more cationic or anionic sublattices with a high configuration entropy. [36,37][38][39][40] These high-entropy materials, a form of multi-carbide solid solution, have drawn widespread attention recently due to their vast potential and broad industrial application prospects.However, the intense research on HECCs has primarily focused on their mechanical properties.
The physical properties of HECs, especially SC and topological properties, are still worth exploring.
In this study, HECs of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx(x = 1 and 0.8) with a singlephase NaCl-type structure were prepared by a spark plasma sintering method.We report our discovery and investigation of the HEC superconductors Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx(x = 1 and 0.8), which shows bulk type-Ⅱ SC with Tc about 4.00 K (x = 1) and 2.65 K (x = 0.8), respectively.Considering the extraordinary properties of the HECCs mentioned above, the discovery of SC and topological band structures in these HECCs would make them an excellent platform for novel quantum device fabrication.
PXRD data were taken on the MiniFlex of Rigaku at a scanning rate of 1 o /min.Through Rietveld refinements in Fullprof suit software, lattice parameters were obtained.Chemical composition was estimated with SEM-EDX with an electron acceleration voltage of 20 KV.The temperature-dependent electrical resistivity and magnetic susceptibility and heat capacity, were measured by a physical property measurement system (PPMS, Quantum Design.Inc).The resistance measurements were performed with a four-probe method.The magnetization and heat capacity measurements use small pieces of sample.The high-pressure resistance measurements were performed at the high-pressure station equipped with a diamond anvil cell at Synergetic Extreme Condition User Facility.In the measurements, the standard fourprobe electrodes (platinum foils) were applied to the samples, and the pressure was determined by the ruby fluorescence method. [53]For all resistivity measurements at ambient pressure, platinum wires were connected to the sample with silver paint.
We perform the calculations using the experimental lattice structure parameters.
The projector augmented-wave (PAW) method [58] with a 400 eV plane-wave cutoff energy is employed.For Brillouin zone sampling, a Γ-centered 5 × 5 × 5 k-points mesh within the Monkhorst-Pack scheme is used in the self-consistent process.Convergence criteria for the electronic self-consistent iteration are set to 10 -6 eV.Spin-orbit coupling (SOC) is used in the calculations of electronic band structure properties.distributed.The proportion of each metal element is close (See Figure S3).Note that this method cannot accurately determine its content since carbon has a light mass and is most likely a contaminant in the EDX analysis.Nevertheless, the EDX results showed that the carbon content of the x = 1 sample is higher than the carbon content of the x = 0.8 sample.Figure 1c-d shows the temperature dependencies of resistivity for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx(x = 1 and 0.8) samples.A sharp resistivity drop is observed in both cases, indicating the superconducting transition.The zero-resistivity was achieved at 4.00 K for x = 1 and 2.68 K for x = 0.8.The normal resistivity decreases only slightly with a near temperature independent, similar to that observed in HEA superconductors. [41,42]The residual resistivity ratio (RRR) value for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx(x = 1 and 0.8) samples is close to 1.
Table S1 summarizes all the gathered normal and superconducting parameters for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxsamples.For comparison, the relevant superconducting parameters of previously reported HECCs are also listed in Table S1. [44,45]e low-temperature specific heat measurements under applied magnetic fields of 0 and 5 T were performed to confirm the bulk nature of the SC.The obvious anomaly in the 0 T heat capacity (Figure 4a  , where μ * represents the Coulomb pseudopotential parameter and is typically given a value of 0.13. [46,47]Based on the obtained values, the superconducting parameter λep = 0.49 for x = 1 and λep = 0.46 for x = 0.8.The λep values suggest that Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CxHECs are weak-coupling superconductors.In crystalline materials, electron-phonon coupling is a ubiquitous many-body interaction that drives conventional superconductivity.[50] In the Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxsystem, the electron-phonon coupling strength weakens as the Tc decreases. To further investigate the superconductivity of HEC, we performed the highpressure resistance measurements for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC. Figure 5a shows the typical resistance curves of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC under various pressures up to 82.5 GPa.It is seen that the superconducting transitions of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC subjected to different pressures are sharp, and the zeroresistance state remains present throughout the full range of pressures applied (see Figure 5b).The Tc shows only a slight change from its ambient-pressure value of 4.15 K to 3.95 K at 82.5 GPa.The pressure-dependent Tc for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC is mapped in the phase diagram in Figure 5c.We see a robust SC against high physical pressure in Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC, with a slight Tc variation of 0.3 K within 82.5 GPa.A similar phenomenon was also observed in (TaNb)0.67(HfZrTi)0.33HEA. [51]is makes superconducting HECs also promising candidates for new applications under extreme conditions.
The lattice parameter of the Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cfitted by the Birch-Murnaghan equation of state is 4.484 Å, which is in consistent with the experimental lattice parameter (a = 4.4573(3) Å) (see Figure S4).For simplicity, the lattice parameters of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxare fixed to the experimentally refined lattice constants.The total density of states (TDOS), the local density of states (DOS), and the partial DOS for Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxare shown in Figure 6.In this work, the supercell contains 64 atoms, so that the experimental doping ratio rTi = rZr = rNb = rMo = rTa = 0.2 cannot be obtained.We consider four different atomic arrangements' structural configurations (the doping ratio is equal to 0.1875 for three elements and 0.21875 for the remaining two.) for investigating the influence of the disorder on the electronic properties of the Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx.The overall shape of the averaged TDOS for x = 1 and x = 0.8 are pretty similar, while is quite different near the Fermi level.The TDOS passing through the Fermi level suggests its typical metallic properties (see Figure 6a and Figure 6b).The local DOS diagram shows that the Ti, Zr, Nb, Mo, and Ta atoms are the most significant contribution to TDOS near the Fermi level.In contrast, the contribution from the C atoms is relatively modest.The d orbital of M and p orbital of C electrons are highly hybridized below the Fermi level.As displayed in Figure 6cf, the projected DOS with angular momentum reveals that the d-electrons of M elements are the main contributions, i.e., 3d for Ti, 4d for Zr, Nb, Mo, and 5d for Ta.These results indicate that the superconductivity may mainly originate from the d-electrons of Ti, Zr, Nb, Mo, and Ta.

Figure
Figure 1a exhibits the powder X-ray diffraction (PXRD) data of the is the geometric parameters of the cuboid sample.The theoretical N values are calculated to be 0.67 (x = 1) and 0.43 (x = 0.8), respectively, consistent with the actual values.The resulting diamagnetic signal with a clear transition to a superconducting state is close to 100 % Meissner volume fraction, indicating the bulk nature of SC in Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxsamples.The onset diamagnetic transition temperatures are 4.00 K for x = 1 and 2.65 K for x = 0.8, which agree well with that from resistivity data.

Figure 7
Figure 7 shows the electronic band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C(x = 1).We first study the band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cwithout SOC.The three different structures are displayed in Figure S5 of Supporting Information.As is shown in Figure 7a-c, there exist six linear band intersections along G-X, G-Y, and G-Z directions at about -0.75 eV (Type -II Dirac-like points (DPs) are denoted by the black circle rectangles).As shown in Figure 7d-e, the linear band intersection along G-X, G-Y, and G-Z directions is not split by considering the SOC, while three linear band intersections along the G-X1, G-Y1, and G-Z1 directions are lightly split (denoted by green circles).The projected band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cshow that the DPs mainly contribute from the d orbitals of transition metals M and the p orbitals of C. As shown in Figure 7h, the positions of the DPs are sensitive to the strain.Therefore, we propose that the Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C is a topological superconductor candidate.Compared with the case of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C, the trivial electronic band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C0.8 are shown in Figure S6.
In conclusion, we have reported synthesized and characterized new HEC superconductors Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cx.Both Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C(x = 1) and Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C0.8 (x = 0.8) are discovered to be bulk superconductors with Tc values of 4.00 and 2.65 K, respectively.The derived superconducting parameters show that Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2Cxare type-Ⅱ BCS weak-coupling superconductors.We observed a robust SC against high physical pressure in Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2CHEC, with a slight Tc variation of 0.3 K within 82.5 GPa.The first-principles calculations show that the DPs exist in the electronic band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C.And the DPs are mainly contributed by the p orbitals of C and the d orbitals of transition metals M. The research results not only expand the new physical properties of HECCs but also provide a new material platform for studying the coupling between SC and topological physics.

Figure 7 .
Figure 7. Electronic band structures of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C(x = 1) calculated by (a)(b)(c) ignoring and by (d)(e)(f) considering the spin-orbit coupling (SOC).The black rectangles indicate the type-II DPs.The three representative crystal structures are used in (a)-(c), shown in Figure S4.(g) The first Brillouin zone of Ti0.2Zr0.2Nb0.2Mo0.2Ta0.2C.(h) Strain dependence of the relative energy at the position of type-II DPs.The Fermi level is indicated by the gray dashed lines.An amplified scaling factor of five is used for the C element in (a)-(f).

Figure S4
Figure S4The calculated total energy as a function of volume.

Figure
Figure S5 (a)-(c) Crystal structures of the three representative structures for Fig. 7(a)-(c).

Table S1
The superconducting parameters of HECCs.