On the basis of the current theoretical understanding of boron-based hard superconductors under ambient conditions, numerous studies have been conducted with the aim of developing superconducting materials with favorable mechanical properties using boron-rich compounds. In this paper, first-principles calculations reveal the existence of an unprecedented family of tetragonal pentaborides MB5 (M = Na, K, Rb, Ca, Sr, Ba, Sc, and Y), comprising B20 cages and centered metal atoms acting as stabilizers and electron donors to the boron sublattice. These compounds exhibit both superconductivity and high hardness, with the maximum superconducting transition temperature Tc of 18.6 K being achieved in RbB5 and the peak Vickers hardness Hv of 35.1 GPa being achieved in KB5 at 1 atm. The combination of these properties is particularly evident in KB5, RbB5, and BaB5, with Tc values of ∼14.7, 18.6, and 16.3 K and Hv values of ∼35.1, 32.4, and 33.8 GPa, respectively. The results presented here reveal that pentaborides can provide a framework for exploring and designing novel superconducting materials with favorable hardness at achievable pressures and even under ambient conditions.
Superconducting and superhard substances have been at the forefront of fundamental research and industrial applications. High-pressure syntheses and theoretical predictions1–12 have been performed to investigate the superconductivity of various materials such as copper/iron-based superconductors,13,14 as well as conventional Bardeen–Cooper–Schrieffer (BCS) superconductors. In view of the tremendous advances that have been achieved in enhancing the transition temperatures of conventional phonon-mediated superconductors, especially of highly compressed hydrides, it is natural to investigate superconductivity in other lightweight materials. This class of compounds containing light elements are excellent candidates for superhard materials.15
Borides have attracted considerable attention in research into superconductivity and hardness because of the diversity of their crystal chemistry.16 In early work, Japanese scientists reported superconductivity of MgB2 at 39 K under ambient pressure,1 fueling enthusiasm for a search for high-temperature superconductivity in the compressed borides.17–24 Polycrystalline MgB2 synthesized by high-pressure sintering is a superhard material25 with Vickers hardness Hv up to 4109.5 kg/cm2. Thus, the idea of developing novel superhard materials using borides and with desired characteristics has gained traction.18,24,26–28 B6C is a superconducting and superhard material, with a transition temperature Tc of ∼12.5 K and an Hv of ∼48 GPa at ambient pressure.29 Theoretical studies30 have revealed that α-BeB6 is superconducting and superhard with Tc of 9 K and Hv of 46 GPa. At high pressure, the predicted Tc and Hv of β-BeB6 are up to 24 K and 31 GPa, respectively. Du et al.23 performed electron–phonon coupling (EPC) and mechanical calculations demonstrating that ScB6 phases have estimated transition temperatures and hardnesses of ∼2.2–5.8 K and 16.2–27.3 GPa under extreme conditions. Lately, Liu’s group have designed a novel class of clathrate-like superconductors, MB7, with superior hardness, among which the Tc and Hv of KB7 were estimated to be 26.2 K and 22.5 GPa at 1 atm. On the basis of previous results,16 borides tend to be hard or superhard materials, with these properties being derived from the covalent boron sublattice. However, although there have been extensive attempts to find high-temperature boron-based superconductors, theoretical and experimental success is yet to be realized. The lower Tc values of two- or three-dimensional borides compared with those of compressed hydrides has motivated us to perform a systematic investigation in this exciting field with a view to finding a new type of superconducting structure with high hardness.
Yttrium borides form various stoichiometries with excellent thermoelastic, electronic, and mechanical properties, which can be systematically applied in various industries.16,27,31 YB2, YB4, YB6, YB12, and YB66 are stable under ambient conditions, with Hv around 25.3–40.9 GPa.16,32,33 Furthermore, the Tc values of YB6 and YB12 are 1.8–12.8 and 4.7 K.17,34,35 We have performed extensive random structure searches for stable or metastable YB2–12 compounds under high pressures and have discovered a novel superconductor, yttrium pentaboride, with a Tc of 12.3 K. In YB5, B atoms are strongly covalently bonded to each other within B20 cages, with Y atoms occupying the centers of these cages and acting as electron donors. A new group of pentaborides, MB5 (M = Na, K, Rb, Ca, Sr, Ba, and Sc), isomorphic to the predicted YB5, have also been estimated to exhibit high superconducting transition temperatures (Tcmax = 18.6 K) and hardness (Hvmax = 35.1 GPa). This study lays the foundation for exploring new classes of hard boron-based superconductors and provides valuable guidance for further experimental syntheses.
II. COMPUTATIONAL DETAILS
III. RESULTS AND DISCUSSION
We investigated structures under ambient pressure and reproduced the previously known phases of YB2, YB4, YB6, and YB12, confirming the reliability of our predictions. The lattice parameters of the optimized structures are consistent with known experimental findings,16 as presented in Table S1 (supplementary material). The formation enthalpies ΔH of the predicted YB2–12 stoichiometries were calculated with respect to the pure elements at 1 atm, 50 GPa, and 100 GPa [see Fig. 1(a)], from which the convex hull was constructed. The known , Pnnm, and Cmca structures for boron47 and the hcp, dhcp, and P6222 structures for yttrium48 were adopted in their corresponding stable pressure. The results revealed that YB2, YB4, and YB12, as expected, are stable at atmospheric pressure. Cubic YB6 is metastable at ∼31 meV/atom above the convex hull, which does not show that our calculations are unreliable or that the material cannot be synthesized experimentally. The fact that the compound is in a metastable state means that the predicted effect can be observed in actual high-temperature experiments.49 At 50 GPa, the stable configurations remain the same, and I4/mmm–YB6 emerges on the convex hull. Hexagonal YB2 emerges as the most stable compound in the studied pressure ranges all along. With the pressure increased to 100 GPa, the stable structure of YB4 changes to C2/m. Meanwhile, YB12 deviates from the tieline and decomposes into YB6 and B, whereas another unreported boron-rich composition P4/mmm–YB5 is close to the convex hull (∼35 meV). The phonon bands of these structures manifest their dynamic stability, since the no imaginary frequency appears in the Brillouin zone (see Fig. S1, supplementary material). In the following discussion, the novel YB5 structure is the main focus because it has relative high symmetry and is recoverable to ambient pressure.
Yttrium pentaboride adopts a tetragonal P4/mmm space group [see Fig. 1(b)], where B atoms occupy 4n and 1d positions bonding to each other within the B20 cage [see Fig. 1(c)], and Y atoms lies at the crystallographic 1b position occupying the centers of the B20 cages. This structure is similar to that of cubic YB6, in which B atoms constitute a boron octahedral network. The straight chains of boron octahedrons are arranged in a three-dimensional frame. This three-dimensional clathrate-like B20 structure with metal atoms filling the B cages is also found in NaB5, KB5, RbB5, CaB5, SrB5, BaB5, and ScB5 through isostructural metal atom replacement at 100 GPa, with each material showing thermal stability (see Fig. S2, supplementary material). Phonon calculations reveal their dynamic stability at ambient pressure (Fig. S3, supplementary material). The electron localization function (ELF)50 was calculated to map the electron pair probability and understand interatomic interactions. No charge localization appears between boron and metal atoms, providing evidence for M–B ionic bonding (see Fig S4, supplementary material). Bader charge analysis was also performed to calculate the transferred charge between atoms (see Table S2, supplementary material). The metal atoms serve as electron donors to the boron sublattice, which is also supported by the ELF results. The ELF values between B–B bonds in the three-dimensional frameworks of the metal pentaborides are larger than 0.7, implying the formation of covalent bonds. Among them, the ELF between two adjacent boron octahedrons is approximately equal to or even greater than 0.9, which is greater than that between B5 (the B atom shared between adjacent boron octahedrons) and its neighboring atoms, which has a range of 0.7–0.8. The integrated crystal orbital Hamilton population (ICOHP) of adjacent B–B pairs with bond lengths of 1.6–2.0 Å was calculated using the LOBSTER package51 to further confirm their covalent characteristics. The negative ICOHP values of B–B pairs in the above structures indicate B–B covalent interactions. As displayed in Fig. S5 (supplementary material), the ICOHP values of B1–B2 (B3–B4) pairs in MB5 are in the range of −7.8 to −5.4 eV/pair, which indicates strong covalent bonding. The B1–B3 (B1–B4, B2–B3 or B2–B4) bonds are fairly strong, with ICOHP values of −5.4 to −2.7 eV/pair, stronger than or approximately equal to that of B5–B1 (B5–B2, B5–B3, or B5–B4, −3.6 to −2.9 eV/pair) bonds. The ICOHP values are consistent with the ELF results.
A high electronic density of states (DOS) is a guideline for successful searches for new superconducting materials, and therefore the projected DOS of MB5 at atmospheric pressure was calculated. Note that all the studied compounds are metallic, as presented in Fig. 2. In the alkali metal (AMB5) and the light alkaline earth metal (AEMB5) pentaborides, the greatest contribution to the DOS close to the Fermi energy Ef is provided by the B-p orbital [see Figs. 2(a)–2(e)], although the M-d orbital at Ef does play a non-negligible role in their metallicity. (It should also be mentioned that the electronic DOS of alkali metal pentaborides has a van Hove singularity near Ef, implying fairly strong electron–phonon interactions associated with boron phonon modes.) By contrast, in BaB5 and the rare-earth borides ScB5 and YB5, the DOS values are dominated by both the M-d and B-p orbitals [see Figs. 2(f)–2(h)]. In addition, ScB5 exhibits a particularly high DOS at the Fermi level (see Fig. S6, supplementary material).
The conclusions regarding the electronic DOS are further confirmed by the results for electron–phonon coupling (EPC). The calculated Eliashberg spectral function α2F(ω) and the EPC parameter λ(ω) at ambient pressure are shown in Fig. 3. Clearly, EPC in the AMB5, BaB5, and ScB5 is much more prominent than in the light AEMB5 and YB5. The calculated EPC values, logarithmic average phonon frequency ωlog, electronic DOS near the Fermi energy N(Ef), and superconducting transition temperatures Tc are presented in Table I. The estimated λ values of NaB5 and KB5 are 0.73 and 0.64, respectively, and the corresponding values of Tc are 17.5 and 14.7 K when the Coulomb potential parameter μ* equals 0.1. According to the phonon DOS (PHDOS) illustrated in Figs. 3(a) and 3(b), vibrations can be separated into low-frequency and high-frequency regions. The low-frequency modes (<8 THz) contribute 52% and 35%, respectively, comprising a mixture of vibrations from M (M = Na and K) and B atoms. EPC calculation for RbB5 gives a relatively large λ of 0.78, which is primarily associated with the high-lying frequencies (>6 THz) of boron (∼63%) [see Fig. 3(c)], yielding an estimated Tc of 18.6 K. The estimated Tc values for CaB5 and SrB5 are 6.6 and 6.8 K, respectively. For the calculated λ of 0.49 and 0.50, as presented in Figs. 3(d) and 3(e), the high-frequency B vibrational modes contribute about 88% and 90%, respectively. The high ωlog and N(Ef) values cannot considerably improve Tc, whereas the weak electron–phonon interaction limits the superconductivity of these compounds. In BaB5, ωlog is slightly lower than in the CaB5 and SrB5 phases. However, Tc reaches 16.3 K, flowing from the higher λ of 0.73. The contribution of 88% to the total EPC is closely related to the high-frequency vibrations. As can be seen, although ScB5 has a stronger EPC interaction and higher N(Ef) than the other pentaborides, its ωlog is quite low, which is caused by λ and is not favorable to Tc. The estimated superconducting transition temperature is ∼14.2 K. YB5 is also predicted to be superconducting, with a Tc of 12.3 K, and corresponding ωlog and λ of 403.8 K and 0.66, respectively. We note that the Tc values of borides are not strongly dependent on the contributions of the DOS of boron atoms to the total DOS at the Fermi energy or the electron–phonon interactions of boron to the total λ.
|Boride .||λ .||ωlog .||N(Ef) (states Ry−1 f.u.−1) .||Tc (K) .|
|Boride .||λ .||ωlog .||N(Ef) (states Ry−1 f.u.−1) .||Tc (K) .|
Compared with high-temperature hydrogen-based superconductors, it is obvious that the EPC parameter λ and ωlog of the borides are much lower owing to the higher mass of boron. However, only a limited difference is observed in N(Ef), a manifestation of the fact that it is the low values of λ and ωlog that lie at the root of the poorer superconductivity of the borides. However, borides can be stabilized at attainable pressures or even ambient pressure, indicating their higher application potential.
|Borides .||C11 .||C33 .||C44 .||C66 .||C12 .||C13 .||B .||G .||Y .||ν .||Hv .|
|Borides .||C11 .||C33 .||C44 .||C66 .||C12 .||C13 .||B .||G .||Y .||ν .||Hv .|
Our extensive structure searches combined with first-principles calculations have revealed the appearance of a metastable clathrate-like B20 structure in tetragonal MB5 (M = Na, K, Rb, Ca, Sr, Ba, Sc, and Y), adopting P4/mmm symmetry. In this structure, metal atoms act as donors to transfer charges to B, and B atoms form a three-dimensional boron octahedral covalent network. Density of states calculations show that the pentaborides are metallic and that the electronic states occupying the Fermi level come mainly from boron atoms. Calculated electron–phonon couplings and elastic constants indicate that the cage-like B20 structure can be viewed as a structural model for hard superconductors, with an estimated highest Tc of 18.6 K and a maximum Hv of 35.1 GPa at 1 atm. It is worth mentioning that KB5, RbB5 and BaB5 are superconducting and hard materials, in which the corresponding superconducting critical temperatures are estimated to be around 14.7, 18.6, and 16.3 K and the Vickers hardness to be 35.1, 32.4, and 33.8 GPa, respectively. Our work provides a reference for future experiments and demonstrates that metastable borides hold considerable promise as superconductors with superior hardness at achievable pressures.
See the supplementary material for supplementary figures and tables.
We thank Dr. Yunxian Liu and Dr. Zihao Huo for AIMD calculations and many stimulating discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12104127 and 22131006), the Doctoral Starting Up Foundation of Hebei Normal University for Nationalities (Grant No. DR2020001), the Clean Energy (Carbon Peaking and Carbon Neutrality) Industry Research Institute of Chengde (Grant No. 202205B090), and the Natural Science Foundation of Shandong Province (Grant No. ZR2020QA060). Parts of the calculations were performed at the High Performance Computing Center of Shandong Bailing Cloud Computing Co., Ltd.
Conflict of Interest
The authors have no conflicts to disclose.
Hui Xie: Conceptualization (equal); Data curation (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Hong Wang: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Fang Qin: Investigation (equal). Wei Han: Investigation (equal). Suxin Wang: Investigation (equal). Youchun Wang: Conceptualization (equal); Data curation (equal); Investigation (equal). Fubo Tian: Software (equal); Writing – review & editing (equal). Defang Duan: Conceptualization (equal); Investigation (equal); Software (equal); Writing – review & editing (equal).
The data that support the findings of this study are available within the article and its supplemental material and from the corresponding authors upon reasonable request.