Superconductivity Above 100 K Predicted in Carbon‐Cage Network

Abstract

To explore carbide superconductors with higher transition temperature, two novel carbon structures of cage‐network are designed and their superconductivity is studied by doping metals. MC₆ and MC₁₀ are respectively identified as C₂₄ and C₃₂ cage‐network structures. This study finds that both carbon structures drive strong electron–phonon interaction and can exhibit superconductivity above liquid nitrogen temperature. Importantly, the superconducting transition temperatures above 100 K are predicted to be achieved in C₂₄‐cage‐network systems doped by Na, Mg, Al, In, and Tl at ambient pressure, which is far higher than those in graphite, fullerene, and other carbides. Meanwhile, the superconductivity of cage‐network carbides is also found to be sensitive to the electronegativity and concentration of dopant M. The result indicates that the higher transition temperatures can be obtained by optimizing the carbon‐cage‐network structures and the doping conditions. The study suggests that the carbon‐cage‐network structure is a direction to explore high‐temperature superconducting carbides.

The novel carbon structures of cage‐network is designed. The superconducting transition temperature above 100 K are predicted to be achieved in C₂₄‐cage‐network systems doped by metals at ambient pressure, which is far higher than those in graphite, fullerene, and other carbides. The study suggests that the carbon‐cage‐network structure is a direction to explore high‐temperature superconducting carbides.

1. Introduction

Superconductivity has always been one of the most fascinating research topics in condensed matter physics. Recently, near room‐temperature superconductivity has been observed in metal hydride at high pressure above 100 GPa,^{[ 1} , ^{2 ]} which attracts much attention to the research field of light‐weight element high‐temperature superconductors. Compared with hydrides, carbides are another widely concerned materials. As we all know, carbon is a magical element and widely exists in our life. With the different hybridization forms of sp ³, sp ², and sp between carbon atoms, different structures and morphologies can be formed such as diamond, graphite, nanotube, and fullerene, etc. Significantly, carbon‐based materials were also suggested to be one of the most promising high‐temperature or room‐temperature superconductors,^{[ 3 ]} which can play important roles in energy, information, electronics, medical, and other fields. Superconductivities of different carbon structures have been explored experimentally and theoretically. However, the superconducting transition temperature (T _c ) of carbide is far from the expected value. For example, the T _c in boron‐doped diamond is only about 4 K,^{[ 4 ]} although the boron content increases, the T _c can reach about 55 K.^{[ 5 ]} For carbon structure with layered characteristics, the highest T _c of metal inserted graphite is 11.5 K (CaC₆).^{[ 6 ]} Recently, the unconventional superconductivity in magic‐angle graphene superlattices has rekindled interest of carbon‐based materials,^{[ 7} , ^{8 ]} but the T _c from the resistance measurement is still below a few Kelvins. In Li‐decorated monolayer graphene, the T _c is only characterized as 5.9 K.^{[ 9 ]} For 1D system, single‐walled carbon nanotube (SWCNT) with a length of 4 Å exhibits the superconductivity of 15 K.^{[ 10 ]} And the T _c in boron‐doped SWCNT is only 12 K.^{[ 11 ]} For fullerene (C₆₀) crystal, the superconductivity was discovered and the T _c can be tuned in the range of 18 − 40 K in alkali metal‐doped C₆₀.^{[ 12} , ^{13 ]} Some clathrate carbon structures similar to C₆₀ were also investigated. For instance, the T _c can be reached 55 K in Na‐doped C₂₀.^{[ 14 ]} And it was predicted that the superconductivity is decreased with the increase of the size of carbon cage.^{[ 15 ]} In a carbide CrC with the rocksalt structure, the superconductivity of 35 K was predicted.^{[ 16 ]} Compared with the highly concerned hydride superconductors nowadays, the T _c of the aforementioned carbides is generally relatively low. However, the superconductivity of these carbides is almost achieved at ambient pressure, unlike hydrides, which often require a pressure of over 100 GPa. Therefore, in exploring the high‐temperature superconductivity at ambient pressure, carbides have more advantages than hydrides. At present, the main reason for the low T _c in carbides is that the optimal crystal structure formed by carbon atoms has not been found. For example, the near room‐temperature superconductivity of hydrides is almost only present in some special structures, such as cage configuration.^{[ 17} , ^{18 ]} Hence, it is very important to explore novel structures for the high‐temperature superconductivity of carbides.

From previous studies, the T _c of carbide is usually <50 K. By investigating the structural characteristics, we can see that C₆₀ molecules are isolated in the fullerene crystal, carbon nanotubes are isolated from each other in the plane direction, and the coupling of graphene in the interlayer direction is prohibited. So we speculate that one of possible reasons of the low T _c is the lack of strong structural coupling between carbon structural units, which leads to relatively low superfluid density. Namely, the strong structural coupling may drive the superconductivity with higher T _c . In a Q‐carbon structure, the superconductivity of T _c = 55 K was observed.^{[ 19} , ^{20 ]} In the clathrate carbon structure of FC₃₄ with the coalescence feature of big and small cages, the T _c was theoretically predicted as 77 K.^{[ 21 ]} The predicted T _c can be close to 80 K in metal‐doped (CB)₃ clathrates.^{[ 22} , ²³ , ^{24 ]} Additionally, in clathrate carbides with sodalite structure, the T _c was predicted to exceed 100 K.^{[ 25} , ²⁶ , ^{27 ]} These reported results imply the importance of structural bond coupling among carbon cluster units. Furthermore, it was found that the clathrate structure is a good material gene and more likely to produce the high superconducting transition temperature in carbides. Of course, it is better to have strong coupling between cages rather than isolated like C₆₀. Here, we design two carbon cages of C₂₄ and C₃₂ by referring to the clathrate structures formed by H atoms in CaH₆ ^{[ 28 ]} and LaH₁₀.^{[ 17 ]} The basic principle of structural design is that carbon atoms form obvious cage‐like features, and there are obvious coupling characteristics between the cages, such as through C‐C bonds or shared atomic surfaces. Under the symmetry of the space group, the carbon‐cage‐network structure‐based on C₂₄ or C₃₂ cage is formed, respectively. The superconductivity above 100 K is predicted in the carbon‐cage‐network structures.

2. Results and Discussion

Figure  1 shows the geometric characteristics of two carbon structures‐doped by metal. Their chemical formulas can be expressed as MC₆ and MC₁₀ respectively, where M represents the doped metal. As shown in Figure 1a, MC₆ possesses the sodalite‐structure with $Im\overline{3}m$ space‐group. The unit cell of MC₆ contains two metal atoms and twelve carbon atoms. A cage made of 24 carbon atoms is formed in MC₆, calling as C₂₄ cage, and it is composed of six quadrilaterals and eight hexagons. Remarkably, seen from the aspect of supercell, these C₂₄ cages are connected by sharing quadrilateral rings in each axial direction, while these cages are overlapping along the diagonal direction of the plane, such as those marked in green and yellow lines in Figure 1b. As a result, these carbon cages forms a 3D network, with doping metal in the center of each cage. This kind of carbon‐cage network in the whole crystal is not only different from the isolated feature of cage in fullerenes^{[ 12} , ^{13 ]} and some so‐called smaller fullerenes such as C₂₀,^{[ 14 ]} C₂₈,^{[ 29 ]} and C₃₆,^{[ 30 ]} but also different from the coalescence of big and small cages in FC₃₄,^{[ 21 ]} LiC₄₀,^{[ 31 ]} and Li₈C₄₆.^{[ 32 ]} For MC₁₀ as shown in Figure 1d, it is another kind of sodalite‐like structure with $Fm\overline{3}m$ space‐group. The unit cell of MC₁₀ consists of four metal atoms and forty carbon atoms, in which a cage made of 32 carbon atoms is formed, calling as C₃₂ cage. Each C₃₂ cage is composed of six quadrilaterals and twelve hexagons. Similar to MC₆, MC₁₀ also exhibits the feature of carbon‐cage network as shown in Figure 1e. But C₃₂ cages are connected by sharing hexagon rings along the diagonal direction of the plane, such as those marked in green and yellow lines in Figure 1e, and these C₃₂ cages are overlapping in each axial direction. Additionally, the viewing from the aspect shown in Figure 1c,f displays the difference between MC₆ and MC₁₀. Comparing C₂₄ with C₃₂ cage, we find that the diameter of the former is smaller than that of the latter. In C₂₄ cage, all the lengths of the nearest neighbor C‐C bonds are uniform, while in C₃₂ cage, there are two kinds of the lengths of the nearest neighbor C‐C bonds. The C‐C bonding length forming hexagonal ring is less than that forming quadrilateral ring in C₃₂ cage.

Figure 1.

The sketch of carbon‐cage‐network doped by metal. a–c) are corresponding to MC₆ with $Im\overline{3}m$ space‐group, while (d–f) are corresponding to MC₁₀ with $Fm\overline{3}m$ space‐group. (a) and (d) is the unit cell of MC₆ and MC₁₀, respectively. Others are supercell configurations.

Based on C₂₄ and C₃₂ cage‐network structures, we have considered the situation that most of the metals in the periodic table are doped into these two carbon structures. The structural optimization, electronic structures, and phonon spectra were calculated based on the first‐principles method. When the doping of a metal satisfies both metallization and dynamical stability, we consider it a candidate system for investigating superconductivity. Table S1 (Supporting Information) presents the evaluation of metallization and dynamical stability for metal‐doped two carbides with cage‐network configuration. To compare with previous predictions,^{[ 25} , ²⁶ , ^{27 ]} the pressure factor was also considered in our calculations. It is found that two cage‐network structures doped by metals are good candidates. The density of states (DOS) shown in Figures S1– S29 (Supporting Information) and phonon spectra without imaginary frequency in Figures S30– S58 (Supporting Information) demonstrate the metallization and dynamical stability of these two cage‐network structures of MC₆ and MC₁₀ doped by different metals at different pressures, respectively.

For stable and metallic phases of MC₆ and MC₁₀, the Eliashberg electron–phonon spectral function α² F(ω) was calculated, and then the electron–phonon coupling (EPC) constant λ, the logarithmic average of phonon frequency ω_log, and the superconducting transition temperature T _c were obtained, which are summarized in Table S2 (Supporting Information). Figure  2 shows the dependence of T _c on pressure for two cage‐network carbides doped by different metals. It is found that many metal dopants lead to the superconductivity of these two cage‐network carbides, and higher T _c can be obtained in MC₆ than MC₁₀. Metal doping makes these two carbon structures exhibit the variety of superconductivity. Remarkably, T _c of MC₆ is predicted to exceed 100 K, when the dopant is Li, Na, K, Mg, Ca, Al, In, Tl, and Pb. As a comparison, T _c of MC₁₀ can also be above 77 K (temperature of liquid nitrogen), when the dopant is Cs, Ba, and La. With regard to pressure effect, some superconducting transition temperatures increase with the increase of pressure, some decrease with the increase of pressure, and the others first increase and then decrease with the increase of pressure. For examples, T _c of MgC₆ increases with the increase of pressure from 111.6 K at ambient pressure to 124.2 K at 85 GPa, T _c of CaC₆ decreases with the increase of pressure from 131.6 K at 145 GPa to 116.0 K at 200 GPa, T _c of CsC₁₀ reaches a maximum of 86.6 K at 10 GPa. By analyzing the electronic structures and phonon characteristics, we found that the influence of pressure on superconductivity is mainly achieved by softening or hardening the phonon modes. For examples, in MgC₆, pressurization will further soften some phonon modes, such as phonon modes near the H k‐point and in the N − Γ k‐path direction (See Figure S35, Supporting Information). On the contrary, in CaH₆, pressurization will further harden some phonon modes, such as phonon modes in the N − Γ k‐path direction (See Figure S36, Supporting Information). However, the average pressure coefficient η = dT _c /dP is relatively small. The absolute η of NaC₆, MgC₆, AlC₆, GaC₆, InC₆, GeC₆, CsC₁₀, and ScC₁₀ is respectively 0.08, 0.15, 0.07, 0.18, 0.09, 0.08, 0.10, and 0.18 KGPa^‐1, which are lower than those of clathrate compounds such as 0.33 of CaH₆,^{[ 28 ]} 0.37 of LaH₁₀,^{[ 33 ]} 0.34 of TbH₁₀,^{[ 34 ]} 0.25 of Al(BN)₃.^{[ 35 ]} The result indicates that the high‐temperature superconductivity of cage‐network carbides produced at high pressure can be well maintained at zero pressure. The superconductivity of NaC₆, AlC₆, GaC₆, and GeC₆ can be stabilized at ambient pressure, which is different from those of Khan et al. ^{[ 26 ]} and Sano et al.,^{[ 27 ]} but supports the results of Lu et al. ^{[ 25 ]}

Figure 2.

Superconducting transition temperature T _c dependence on pressure. (a) and (b) are corresponding to MC₆ and MC₁₀, respectively. The results are based on µ^⋆ = 0.1.

As a novel carbon structure, the superconductivity at ambient pressure is still of great interest. Figure 2 shows that NaC₆, MgC₆, AlC₆, GaC₆, InC₆, GeC₆, CsC₁₀, and ScC₁₀ can exhibit the high‐temperature superconductivity at ambient pressure. In particular, T _c of NaC₆, MgC₆, AlC₆, InC₆, and TlC₆ respectively is predicted as 113.3, 111.6, 110.7, 104.6, and 104.0 K at ambient pressure. As shown in Figure  3 , the predicted T _c 's of MC₆ and MC₁₀ are much >9.8 K of CaC₂,^{[ 36 ]} 11.2 K of MgC₂,^{[ 37 ]} 11.5 K of NbC,^{[ 38 ]} 15 K of PdC,^{[ 39 ]} and 34.6 K of CrC^{[ 16 ]} without cage structure. Importantly, the predicted T _c above 100 K for MC₆ at ambient pressure is obviously higher than those of low dimensional carbon structures such as 4 and 55 K of boron‐doped diamond,^{[ 4} , ^{5 ]} 12.4 K of strained‐diamond,^{[ 40 ]} 39 K of hybridized graphite‐diamond,^{[ 41 ]} 11.5 K of graphite‐like CaC₆,^{[ 6 ]} 5.9 K of Li‐doped graphene,^{[ 9 ]} 15 K of SWCNT,^{[ 10 ]} and 12 K of boron‐doped SWCNT.^{[ 11 ]} It is found that our predicted T _c of >100 K in MC₆ is also much higher than the superconducting transition temperature of those carbon structures with isolated carbon cages such as 11 K of T‐carbon,^{[ 42 ]} 18–38 K of alkali metal‐doped C₆₀,^{[ 43} , ⁴⁴ , ⁴⁵ , ^{46 ]} and 55 K of NaC₂₂.^{[ 14 ]} This improvement in superconductivity is attributed to this novel carbon structure of cage‐network. With the same cage‐network feature, MC₆ can product higher T _c than Q‐carbon,^{[ 19 ]} FC₃₄,^{[ 21 ]} and metal‐doped (CB)₃ clathrates.^{[ 22} , ²³ , ^{24 ]}

Figure 3.

Comparison of T _c values of carbon‐based materials at ambient pressure. Our results are marked with the red half full symbols. The solid symbol represents the experimental results, while the hollow symbol represents the theoretical ones.

Pure cage‐network structure is an insulator, for example, the bandgap of C₂₄‐cage‐network is about 2.5 eV,^{[ 27 ]} while the metal doping drives the transition from insulator to metal. This driven force is mainly from the charge transfer from metal to cage‐network (see Figure S59, Supporting Information). Taking MgC₆ at 0 GPa as representative of MC₆, the transferred charge from Mg to C₂₄‐cage‐network is about 1.48 eMg^‐1. The electron localization function (ELF) shown in Figure  4b also displays the lack of covalent bond between Mg and C. The electrons from the metal enter the C‐2p orbit and are transmitted in the network of carbon cages, which causes the empty occupied conduction bands to move to the lower energy level. As a result, three energy bands cross over the Fermi level of MgC₆, forming the complicated Fermi surface (FS) sheets including hole‐ and electron‐like and exhibiting the metallic feature as shown in Figure 4a,c. The electronic states near the Fermi level are contributed by C‐2s and 2p. The DOS value at Fermi level is 1.73 states/eV/f.u. (or 0.25 states/eV/atom) for MgC₆. As a comparison, in the metal‐doped C₆₀ with fcc group‐space, there are also several energy bands crossing the Fermi level, but the DOS value at Fermi level is only about 0.10 − 0.13 states/eV/atom.^{[ 47} , ^{48 ]} In NaC₂₂ where carbon cage is also isolated, the DOS value at Fermi level is about 0.2 states/eV/atom,^{[ 14 ]} <0.25 states/eV/atom of MgC₆. Additionally, the cage‐network structure results in the wider energy band near Fermi level than the cage‐isolated structure, such as about 5 eV for MgC₆ comparing with about 0.5 eV for K‐doped C₆₀ ^{[ 47 ]} and 1.9 eV for NaC₂₂.^{[ 14 ]} The high electronic DOS value at Fermi level caused by the cage‐network structure should be one of the reasons for the high‐T _c superconductivity of MC₆ than cage‐isolated structure. Furthermore, in MC₆, different metal doping leads to the difference of electronic states as shown in Figures S1– S29 (Supporting Information), which is one of the reasons why they exhibit different superconductivity.

Figure 4.

Electronic structures, phonon characteristics, and electron‐phonon coupling of MgC₆ at 0 GPa. a) Electronic energy band structure along high‐symmetrical k‐point path shown in (c) and total and projected density of states (DOS) on atomic orbitals. b) Electron localization function. c) Fermi surface features in the first Brillouin zone (FBZ) corresponding to three energy bands crossing the Fermi level. d) Phonon spectra, phonon density of states (PhDOS) projected on atoms, Eliashberg spectral function α² F(ω), and electron–phonon coupling integral λ(ω).

As shown in Figures S30– S47 (Supporting Information), compared with metal‐doped C₆₀, the highest phonon vibration frequency of C atoms in MC₆ is greatly reduced at ambient pressure, almost from about 1600 cm⁻¹ of metal‐doped C₆₀ ^{[ 48 ]} to 900 cm⁻¹ of the MC₆. For example, the highest phonon frequency in MgC₆ is 833 m⁻¹ as shown in Figure 4d. However, the cage‐network structures result in the stronger EPC constants (See Table S2, Supporting Information) than that of cage‐isolated structure. As shown in Figure 4d, the EPC constant of MgC₆ at ambient pressure is λ = 2.88, which is far >0.4 − 0.9 of metal‐doped C₆₀ ^{[ 48 ]} and 1.12 of NaC₂₂.^{[ 14 ]} Such a strong electron–phonon interaction mainly comes from the contribution of C phonon modes, and the strong EPC is a main factor for the high superconductivity of the cage‐network structure.

For MC₁₀, only CsC₁₀ and ScC₁₀ are superconducting at ambient pressure, the T _c are 74.3 and 58.1 K, respectively. From the perspective of comparing electronic states and phonon structures, the reasons for the different T _c of MC₆ and MC₁₀ was also analyzed. In the case of CsC₁₀ at ambient pressure shown in Figure  5 , on one hand, the transferred charge is about 0.76 eCs^‐1, only one band crosses over the Fermi level forming hole‐ and electron‐like FS sheets. However, the DOS value at Fermi level of CsC₁₀ is 0.30 states/eV/atom, slightly >0.25 states/eV/atom of MgC₆. On the other hand, the higher phonon frequency of 1065 cm⁻¹ is obtained in CsC₁₀ than MgC₆. The EPC of MgC₆ is λ = 2.88 with ω_log = 449.7 K, while λ = 2.30 and ω_log = 346.2 K for CsC₁₀. The electron–phonon interaction in CsC₁₀ is slightly weaker than that in MgC₆. Furthermore, the difference between them mainly results from the different α² F(ω) caused by the phonon states of C atoms. As mentioned above, all the lengths of the nearest neighbor C‐C bonds are uniform in MC₆, while in MC₁₀, there are two kinds of the lengths of the nearest neighbor C‐C bonds (See Figure S60, Supporting Information). The C‐C bonding lengths forming hexagonal ring in C₃₂ cage are shortened, which leads to the enhancement of some phonon vibrations, that is, the phonon becomes hard. As a result, the contribution of these phonon modes to the EPC decreases. As shown in Figures 4d and 5d, although the increase of phonon frequency increases the integration range, the phonon states (PhDOS and α² F(ω)) in the low‐frequency region that contributes greatly to the EPC decrease significantly, which leads to the decrease of the final EPC constant.

Figure 5.

Electronic structures, phonon characteristics, and electron–phonon coupling of CsC₁₀ at 0 GPa. a) Electronic energy band structure along high‐symmetrical k‐point path shown in (c) and total and projected density of states (DOS) on atomic orbitals. b) Electron localization function. c) Fermi surface features in the first Brillouin zone (FBZ) corresponding to one energy band crossing the Fermi level. d) Phonon spectra, phonon density of states (PhDOS) projected on atoms, Eliashberg spectral function α² F(ω), and electron–phonon coupling integral λ(ω).

We can see that the trend of T _c decreasing from MC₆ to MC₁₀ is significantly different from that of T _c increasing from MH₆ to MH₉ and then to MH₁₀. From the perspective of electronic structures and phonon spectra characteristics, we have analyzed their differences in detail. For the doping of the same element at the same pressure (See Figures S1– S29, Supporting Information), the DOS value at Fermi level increases from MC₆ to MC₁₀, which is similar to that of from YH₆ to YH₉ and then to YH₁₀.^{[ 18 ]} At the same pressure, the highest frequency of phonon vibration increases with the increase of C content from MC₆ to MC₁₀, and this phenomenon of the highest phonon frequency increasing with the increase of H content was also observed in hydrides such as from LaH₈ to LaH₁₀.^{[ 17 ]} However, the T _c of carbides and hydrides exhibit opposite trends, with the former decreasing with an increase in C content and the latter increasing with an increase in H content. We analyzed the phonon spectrum and found that this reason may mainly come from the decrease of soft phonon modes with the increase of C content in carbides, and the increase of soft phonon modes with the increase of H content in hydrides.^{[ 17} , ^{18 ]} The stronger hybridization between carbon atoms (See Figure S61, Supporting Information) increases their stability at ambient pressure and also makes phonons harder, resulting in a decrease in superconductivity.

Although the doped metal only provides the transferred charge and only contributes about a quarter to the total EPC constant, the different doped metals also cause the diversity of superconductivity. By analyzing the relationship between different metals and transition temperature, we found some simple rules. Focusing on those superconductors at ambient pressure combining with previous studies,^{[ 25 ]} as shown in Figure  6 , it is found that the T _c decreases with the increase of Allred‐Rochow electronegativity^{[ 49 ]} and valence state of M in MC₆. Alkali metals and alkaline earth metals exhibit better advantages in exploring the higher T _c in cage‐network carbides, which is just why these metals were mainly used in doping C₆₀ ^{[ 12} , ^{13 ]} or other carbides.^{[ 14} , ³¹ , ^{32 ]} As a result, Figure 6 suggests the direction of exploring high‐T _c clathrate carbide superconductors, toward to metals with low electronegativity and low valence state. This law also seems to be applicable to hydrides with clathrate structure formed by H atoms. High or near room‐temperature superconductivity is often observed in clathrate hydrides‐doped by metal with low electronegativity and valence state. The superconductivity with T _c 's over 200 K was predicted theoretically in CaH₆,^{[ 28 ]} YH₆,^{[ 50 ]} MgH₆,^{[ 51 ]} LaH₁₀,^{[ 17 ]} YH₁₀,^{[ 17} , ^{18 ]} ThH₁₀,^{[ 52 ]} AcH₁₀,^{[ 53 ]} TbH₁₀,^{[ 54 ]} CaYH₁₂,^{[ 55 ]} and Li₂MgH₁₆,^{[ 56 ]} respectively. Metals such as La, Y, Li, Mg, and Ca have relatively weak electronegativity, and their doping with hydrides leads to higher T _c . In addition, we have investigated the influence of dopant content on T _c . Changing the doping concentration, the T _c of MC₆ which can be stable at ambient was calculated. As listed in Table S3 (Supporting Information), the metal content in MC₆ is reduced, and T _c is significantly reduced. The result shows that the superconducting transition temperature is sensitive to the metal doping concentration, which means that we can obtain higher T _c by adjusting the metal content. For defect effects, Figures S62 and S63 (Supporting Information) show the defect structures of C atomic vacancy and doping charge and phonon spectra with these defects, taking NaC₆ and CsC₁₀ at 0 GPa as examples, respectively. The results indicate that the systems are dynamically unstable under defects. Hence, we speculate that attempting to regulate superconductivity by adjusting defects in MC₆ and MC₁₀, such as C vacancies and doping charges, is not a very effective means.

Figure 6.

Phase diagram of T _c and dopants. The dependence of T _c on the electronegativity and valence states of dopant for MC₆ superconductor at ambient pressure. The contour colors imply the range of T _c values. Results of non‐metal doping are taken from previous reports.^{[ 25 ]}

Finally, the mechanical and thermodynamic stabilities and the possibility of experimental synthesis of these two kinds of cage‐network carbides were simply analyzed. The mechanical stability was analyzed by calculating the elastic constants of MC₆ and MC₁₀, especially for systems that can superconduct at ambient pressure (See Table S4, Supporting Information). The results show that both MC₆ and MC₁₀ meet the criteria of elastic stability, which means that these cage‐network structures are mechanically stable. The enthalpy of formation were calculated within the quasi‐harmonic approximations (QHA)^{[ 57 ]} when the possibly synthesizing routes of metal + diamond and metal + graphite were assumed. Considering the pressure range of 0 − 200 GPa and the temperature range of 0 − 2000 K (Figures S64 and S65, Supporting Information), it is found that the enthalpy of formation is negative. This means that these two kinds of cage‐network structures are thermodynamically stable and can be synthesized starting from metal and diamond or graphite under certain conditions. The order of magnitude of decomposition enthalpy is in the range of −0.45 − −0.1 eV for MC₆ and MC₁₀. Clathrate hydrides generally exist under higher pressure. Compared with the successfully synthesized hydrides such as LaH₁₀,^{[ 1} , ^{2 ]} YH₆,^{[ 58} , ^{59 ]} and CaH₆,^{[ 60 ]} it was found that the decomposition enthalpy is comparable. For example, the decomposition enthalpies are about ‐0.35 eV at 150 GPa and 0.40 eV at 300 GPa for LaH₁₀,^{[ 17 ]} ‐0.72 eV at 160 GPa for YH₆,^{[ 50 ]} and ‐0.9 eV at 150 GPa for CaH₆,^{[ 28 ]} respectively. This implies the possibility of experimental synthesis of clathrate carbides at a certain temperature and pressure. More synthesis routines have been explored by comparing the enthalpy of formation between MC₆ (or MC₁₀) and metal + C₂₄ cage (or C₃₂ cage), as well comparing the enthalpy of formation of metal carbide + carbon substance (such as graphite or diamond). The results (See Figure S66, Supporting Information) not only show that MC₆ and MC₁₀ have good thermodynamic stability, but also provides some possible synthesis routines. For example, MgC₆ can be synthesized through various forms such as Mg + carbon substance (such as graphite or diamond), MgC + carbon substance, MgC₂ + carbon substance, etc. Actually, some cage‐isolated systems and cage‐coupled materials have been reported to be successfully experimentally synthesized, such as SiB₆,^{[ 61 ]} metal‐doped C₆₀ superconductors,^{[ 12} , ¹³ , ^{62 ]} and carbon–boron clathrate structure.^{[ 63 ]} Zhu et al. suggested a synthetic route of carbon–boron clathrates that metal carbides and borides were mixed and heated to 2500 K and the pressure was controlled in the range of 50 − 150 GPa.^{[ 63 ]} As a reference, clathrate carbides can be synthesized under the similar conditions, especially after removing pressure, these carbides will maintain good stability and superconductivity, as the pressure effect mentioned above. Moreover, the quasi‐hexagonal‐phase fullerene has been successfully synthesized, where the C₆₀ cage‐network was formed via an interlayer bonding.^{[ 64 ]} These experiments also provide a reference for the synthesis of these carbon‐cage‐network structures.

3. Conclusion 

In summary, based on C₂₄ and C₃₂ cages, we have respectively designed two carbon‐cage‐network structures and systematically studied their structural stability, electronic properties, phonon spectra, and electron–phonon interactions after doping metals. The strong electron–phonon interactions were obtained in these two kinds of carbon‐cage‐network. The predicted T _c s in MC₁₀ are slightly lower than those in MC₆. At ambient pressure, the highest T _c induced by C₃₂‐cage‐network structure is 74 K that is obtained in CsC₁₀. Remarkably, C₂₄‐cage‐network structure‐doped by Na, Mg, Al, In, and Tl exhibits the superconductivity above 100 K at ambient pressure, which is far higher than those in graphite, fullerene, and other carbides. The results indicate that the superconductivity of cage‐network carbides is sensitive to the electronegativity and concentration of dopant. Our current results provide a useful route for designing high‐T _c superconductors. And we expect that the work will stimulate further experimental and theoretical studies for exploration of carbide high‐temperature superconductors.

4. Simulation Details

All calculations of structural optimization and electronic structures were carried out by using the density functional theory of Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA)^{[ 65 ]} and the projector augmented wave pseudopotential^{[ 66 ]} as implemented in the Vienna ab initio simulation package (VASP).^{[ 67} , ^{68 ]} The plane wave cut‐off energy was set as 600 eV. The k‐point of the Brillouin zone was 0.03 Å⁻¹ interval distribution of Monkhorst‐Pack for the optimization of structures, and the k‐point interval of the total energy self‐consistent calculation was 0.02 Å⁻¹ or better. Convergence thresholds were set as 10⁻⁵ eV in energy and 10⁻³ eVÅ^‐1 in force.

Within the framework of density functional perturbation theory and PBE‐GGA functional, QUANTUM ESPRESSO package (QE)^{[ 69} , ^{70 ]} was used to calculate the phonon frequency (ω) and the Eliashberg electron–phonon spectral function [α² F(ω)]. Based on α² F(ω), the electron–phonon coupling constant (λ, EPC) of these clathrate compounds was calculated, which is defined by integration over the entire frequency domain of α² F(ω):

$$\lambda =2{\int }_0^{\infty }\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega .$$ (1)

Then T _c was calculated by Allen‐Dynes‐corrected McMillan equation^{[ 71 ]}:

$$T_{\mathrm{c}}=f_1{f}_2\frac{{\omega }_{\log }}{1.2}\exp \left[-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}\right],$$ (2)

where, the factor f ₁ f ₂ is decided by the λ, µ^⋆, ω_log, and mean square frequency ($\overline{{\omega }_2}$),^{[ 71 ]} and the logarithmic average of phonon frequency ω_log is written as:

$${\omega }_{\log }=\exp \left[\frac{2}{\lambda }{\int }_0^{\infty }\frac{{\alpha }^{2}F(\omega )\log (\omega )}{\omega }d\omega )\right].$$ (3)

The typical value of Coulomb pseudopotential µ^⋆ was set as 0.1 for all clathrate carbides. Ultrasoft pseudopotentials for metal and carbon were employed in this calculation. And a k‐mesh of 16 × 16 × 16 in the first Brillouin zone was used in the calculation of the electron–phonon interaction matrix element and a q‐mesh of 4 × 4 × 4 was used for the phonon spectra calculation. The cut‐off energies for wave function and charge density were set as 80 Ry and 600 eV, respectively. At the same time, the forces and stresses of the convergent structure were optimized and controlled within the error range of VASP and QE programs.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Hai Y.‐L., Jiang M.‐J., Tian H.‐L., Zhong G.‐H., Li W.‐J., Yang C.‐L., Chen X.‐J., Lin H.‐Q., Superconductivity Above 100 K Predicted in Carbon‐Cage Network. Adv. Sci. 2023, 10, 2303639. 10.1002/advs.202303639

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.