Prediction of fully metallic $\sigma$-bonded boron framework induced high superconductivity
above 100 K in thermodynamically stable Sr₂B₅ at 40 GPa

The quest for high-temperature superconductors has been the long-sought target since the discovery of superconductivity in solid mercury in 1911 Onnes (1911). According to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, light-element compounds are considered as promising candidates for high superconducting critical temperature ($T_{\mathrm{c}}$) superconductors since they possess high Debye temperature, which is proportional to the $T_{\mathrm{c}}$ Suhl et al. (1959). Recently, the theory-orientated experiments have established near-room-temperature superconductivity in a class of hydrides at high pressure, such as H₃S, LaH₁₀, and CaH₆ Drozdov et al. (2015); Peng et al. (2017); Drozdov et al. (2019); Somayazulu et al. (2019); Ma et al. (2022); Li et al. (2014); Liu et al. (2017); Wang et al. (2012a). Nevertheless, the pressure required to realize these superconducting hydrides is exceptionally high and challenging to achieve. Therefore, it is of significant importance to search for alternative high-$T_{\mathrm{c}}$ superconductors at moderate pressures and even at ambient pressure for practical applications.

At low pressure, MgB₂ is the BCS superconductor with the highest $T_{\mathrm{c}}$ of 39 K owing to its high Debye temperature and strong electron-phonon coupling (EPC), primarily originating from the metalized $\sigma$-bonding bands of in-plane $sp^{2}$-hybridized boron atoms Nagamatsu et al. (2001); Kortus et al. (2001). Thus, metal borides characterized by metallic $\sigma$ bonds have attracted considerable attention in searching for high-$T_{\mathrm{c}}$ superconductors. Besides MgB₂, a class of MB₂, MB₆, and MB₁₂ (M represents transition metals) have also been experimentally identified to be superconducting with $T_{\mathrm{c}}$ below 20 K at ambient pressure Fisk et al. (1971); Schneider et al. (1987); Lortz et al. (2006); Akopov et al. (2017). Recently, Pei et al. discovered that $\alpha$-MoB₂, isostructural with MgB₂, exhibits the second-highest $T_{\mathrm{c}}$ of 32 K at 100 GPa among all known borides Pei et al. (2023). However, as the main contribution at the Fermi level of these borides is not derived from $\sigma$ bonds of boron atoms, none of them have achieved a $T_{\mathrm{c}}$ surpassing that of MgB₂ yet.

In addition to experiments, much effort has also been devoted to theoretical investigations to design borides with high $T_{\mathrm{c}}$. On one hand, thermodynamically stable boride superconductors, such as BeB₆, CaB, and SrB characterized by deformed $sp^{2}$-hybridization Wu et al. (2016); Cui et al. (2022); Shah and Kolmogorov (2013), exhibit lower $T_{\mathrm{c}}$ compared to MgB₂ due to the weaker metallic $\sigma$ bonds, which is similar to the experimental observation. On the other hand, a series of MB₂ monolayer and MB₄ trilayers, such as AlB₂, MgB₄, and InB₄ Zhao et al. (2019, 2020); Wang et al. (2021a), have been predicted to exhibit high $T_{\mathrm{c}}$ due to the strong metallic $\sigma$ bonds originating from $sp^{2}$-hybridized boron atoms, while their producibility is unclear since their thermodynamic stability remains unknown. Similarly, various metastable $sp^{3}$-hybridized borocarbides and boronitrides have also been predicted to exhibit higher $T_{\mathrm{c}}$ than MgB₂ Geng et al. (2023); Gai et al. (2022); Ding et al. (2022); Zhu et al. (2020); Zhang et al. (2022); Wang et al. (2021b), although none of them has been experimentally synthesized to date. Thus, it is encouraging to design metal borides with favorable thermodynamic stability and strong metallic $sp^{2}$/$sp^{3}$ $\sigma$-bonding characteristics that may lead to synthesizable high-temperature superconductors with a $T_{\mathrm{c}}$ higher than MgB₂.

In this work, we performed a comprehensive first-principles structure searches for the alkaline-earth (AE) metal borides (M_xB_y, M = Mg, Ca, Sr, Ba, $x$ = 1-3, $y$ = 1-8) at pressures below 100 GPa via the swarm intelligence-based CALYPSO method Wang et al. (2010, 2012b). Besides the reproduced already known phases Esfahani et al. (2017); Shah and Kolmogorov (2013); Kolmogorov et al. (2012); Cui et al. (2022), we predicted several new thermodynamically stable borides under high pressures. Among them, the $F$$\overline{4}$3$m$ Sr₂B₅ consists of a unique fully $sp^{3}$-hybridized $\sigma$-bonded boron framework, where the density of states at Fermi level is mainly contributed by $\sigma$-bond. This results in a high $T_{\mathrm{c}}$ above 100 K, not only surpassing the boiling temperature (77 K) of liquid nitrogen but also setting a new record for the highest $T_{\mathrm{c}}$ among all known thermodynamically stable boride superconductors.

Our main structure searching results of AE-metal borides M_xB_y below 100 GPa are presented in the convex hull and phase diagrams of Fig. 1 and Fig. S1 SI . Considering the important contribution of zero-point energy (ZPE) to the free energy of compounds containing light elements Hu et al. (2013); Peng et al. (2012), the formation energies of the metal borides were calculated with the inclusion of ZPE. In addition to the known AE-metal borides Esfahani et al. (2017); Shah and Kolmogorov (2013); Kolmogorov et al. (2012); Cui et al. (2022), we uncovered several new thermodynamically stable M_xB_y compounds, including Mg₃B₈, Ca₂B₅, Ca₃B₅, Sr₂B₅, and BaB with $C$2/$m$, $P$6₃/$mmc$, $P$6₃/$mmc$, $F$$\overline{4}$3$m$, and $R$$\overline{3}$$m$ structure, respectively, where Sr₂B₅ further transforms into $P$6₃/$mmc$ phase at 44 GPa (The detailed structural information is listed in Table S1 SI ). The absence of imaginary frequency in the phonon spectra indicates that these borides are also dynamically stable (Fig. S2 SI ), confirming the robustness of the stability. By examining the structures, we found that, except for $F$$\overline{4}$3$m$-Sr₂B₅, all the M_xB_y compounds adopt layered-like morphology with alternatively stacked boron and metal atomic layers (Fig. S3 SI ). The boron atoms should be mainly $sp^{2}$-hybridized in these M_xB_y compounds like other layered metal borides Wu et al. (2016); Cui et al. (2022); Shah and Kolmogorov (2013). On the other hand, $F$$\overline{4}$3$m$-Sr₂B₅ exhibits a unique three-dimensional boron framework as presented in Fig. 2. It consists of two nonequivalent boron atoms, namely B_D and B_T, occupying 4c (0.25, 0.25, 0.25) and 16e (0.59, 0.59, 0.41) Wyckoff position, respectively, where both of them are four coordinated. Specifically, B_D is surrounded by four B_T atoms, which forms a regular tetrahedron B₅ unit similar to C atoms in diamond, indicating B_D is $sp^{3}$-hybridized. B_T has three shorter B_T-B_T bonds and one slightly longer B_T-B_D bond and forms an irregular tetrahedron B${}_{5}^{{}^{\prime}}$ unit, in which B_T atoms also form a regular tetrahedron B₄ unit by itself. This is close to the motif of C atoms in T-carbon Sheng et al. (2011); Zhang et al. (2017), implying B_T exhibits resembling anisotropic $sp^{3}$-hybridization due to the unevenly distributed bond lengths. Further electron localization function (ELF) and crystal orbital Hamiltonian population (COHP) calculations confirm the strong covalent nature between B atoms (Fig. S4 SI ). Meanwhile, Sr-B is ionically bonded since electrons are transferred from Sr to B. Remarkably, the boron sublattice of $F$$\overline{4}$3$m$-Sr₂B₅ can be viewed as isostructure to diamond, where one of the nonequivalent atomic position is occupied by B₄ unit instead of B atom. Thus, to the best of our knowledge, $F$$\overline{4}$3$m$-Sr₂B₅ should be the first metal boride exhibits fully $sp^{3}$-hybridized $\sigma$-bonded boron framework that might be beneficial to the superconductivity.

We next examined the electronic properties of the AE-metal borides by modeling the band structures and density of states (DOS) as shown in Fig. 3 and Fig. S5 SI . Clearly, the newly predicted metal borides are all metallic. Based on the boron hybridization scheme, these borides exhibit different band structure characteristics. The layered-like M_xB_y compounds have both $\sigma$ and $\pi$ bands crossing the Fermi level and $F$$\overline{4}$3$m$-Sr₂B₅ only has $\sigma$ conducting bands, analogous to that of MgB₂ and hole-doped diamond, respectively. Besides the metal-rich Ca₃B₅ and BaB, boron dominates the contribution to the DOS at the Fermi level (N_Ef) due to the charge transfer from metal to boron atoms, as deciphered in Fig. 3b. In the case of metal-rich borides, the amount of electrons provided by the metal atoms exceeds the needs of fully shelled boron atoms. Thus, N_Ef is dominated by metal atoms. Importantly, among these metal borides, $F$$\overline{4}$3$m$ Sr₂B₅ exhibits not only the highest total N_Ef but also the highest boron contribution, which is almost twice as much as that of MgB₂. This is because the Fermi level is located right at the peak position of the DOS, corresponding to a van Hove singularity along the K-$\Gamma$ direction. The projected DOS further reveals that the electrons in the vicinity of the Fermi level are coming from B_T, where B_D merely contributes to the N_Ef because it is nearly fully shelled due to the ideal $sp^{3}$-hybridization. The high N_Ef together with the solely $\sigma$ characteristic of boron suggest $F$$\overline{4}$3$m$ Sr₂B₅ should be a potential high-temperature superconductor if $\sigma$ electronic bands couple strongly to the corresponding phonon modes. We also noticed that Ca₂B₅ has a boron contributed N_Ef higher than that of MgB₂. However, it is a mixture of $\sigma$ and $\pi$ electrons. The superconductivity of Ca₂B₅ might not be able to surpass that of MgB₂.

To investigate the potential superconductivity in these AE-metal borides, electron-phonon coupling simulations were performed. Encouragingly, the estimated EPC parameters, $\lambda$, of $F$$\overline{4}$3$m$ Sr₂B₅ is as high as 3.32 at 40 GPa. This value not only is nearly 3-4 times higher than that of MgB₂ but also higher than that of any $sp^{3}$-hybridized borocarbides studied so far and close to that of high-$T_{\mathrm{c}}$ clathrate superhyrides. As depicted in Fig. 4(a), the large coupling strength is derived primarily from the $T_{2}$ and E modes over the whole Brillouin zone, which corresponded to the stretching of Sr-B bonds (2-7 THz) and rocking vibrations of B₄ units (8-10 THz). The integral ($\lambda$($\omega$)) of the frequency divided Eliashberg phonon spectral function ($\alpha^{2}F(\omega)/\omega$) suggests these two modes contribute almost equally to the overall $\lambda$. This phenomenon is different from MgB₂ and metal borocarbides but similar to $\alpha$-MoB₂ as metal atom shows pronounced contribution to N_Ef. Since the electron-phonon coupling is quite strong in $F$$\overline{4}$3$m$ Sr₂B₅ ($\lambda$ > 1.5), the $T_{\mathrm{c}}$ values and superconducting energy gaps ($\Delta$) were evaluated through the direct numerical solutions of the anisotropic Migdal-Eliashberg equations Eliashberg (1960); Sanna et al. (2018); Giustino et al. (2007) using the calculated logarithmic average frequency ($\omega_{\mathrm{log}}$) and typical Coulomb potential parameters ($\mu$*) of 0.1, 0.13, and 0.16 . The estimated single $\sigma$ gap is about 22 meV (Fig. S8 SI ) and $T_{\mathrm{c}}$ value is 105 K using $\mu$* = 0.13 for $F$$\overline{4}$3$m$ Sr₂B₅ at 40 GPa (Fig. 4), tripling that of MgB₂ and validating the observation of $\lambda$ (those for $\mu$* = 0.1, 0.16 are also provided in Fig.4 and Table S3 SI ). This predicted $T_{\mathrm{c}}$ certainly exceeds the boiling point of liquid-nitrogen (77 K), setting a new superconducting record among all known thermodynamically stable metal borides to the best of our knowledge.

Given the low mass of B, previous studies have demonstrated that anharmonicity played an important role in determining the superconductivity of MgB₂, especially for the $E_{2g}$ phonons Liu et al. (2001); Yildirim et al. (2001); Lazzeri et al. (2003). We then carried out additional EPC calculations to investigate the occurrence of anharmonic effects using the stochastic self-consistent harmonic approximation Errea et al. (2014). As shown in Fig. 4, the anharmonic correction only hardens and broadens the $B_{4}$ rocking E modes, while other phonon modes barely change, similar to that of MgB₂. The average phonon frequency of E modes increases from 9.3 THz to 11.6 THz and the width of E modes is increased by 1 THz. Meanwhile, E modes become the major contributor to the anharmonic Eliashberg function, accounting for 71$\%$ of total $\lambda^{\mathrm{anh}}$, while the contribution of $T_{2}$ modes reduces to 25$\%$, indicating the superconductivity of $F$$\overline{4}$3$m$ Sr₂B₅ should be mainly determined by the coupling between $\sigma$ bands and E phonons. This results in a reduction of the anharmonic EPC parameter $\lambda^{\mathrm{anh}}$ to 2.02, 40$\%$ smaller than the harmonic one. In contrast, the anharmonicity enhances the logarithmic average frequency $\omega_{\mathrm{log}}$ by 34$\%$ from 354 K to 477 K. Consequently, the anharmonic $T_{\mathrm{c}}^{\mathrm{anh}}$ value is dropped by 10$\%$ to 95 K, which still is quite high.

As for other newly predicted borides, only Ca₂B₅, $P$6₃/$mmc$ Sr₂B₅, and BaB exhibit superconductivity, with calculated $T_{\mathrm{c}}$ values of 3.6 K, 6.4 K, and 3.9 K at 30, 50, and 80 GPa, respectively. The details of superconducting properties are shown in Table S4 SI .

The high-temperature superconductivity of $F$$\overline{4}$3$m$ Sr₂B₅ drives us to further evaluate its synthesizability. As shown in Fig. S10 SI , the pressure-temperature phase diagram of Sr₂B₅ was constructed up to 100 GPa and 2000 K via harmonic free-energy calculations since most of the metal borides are synthesized at high temperatures Friedrich et al. (2011). Clearly, once temperature effects are considered, the stable range of $F$$\overline{4}$3$m$ Sr₂B₅ increases from 38-44 GPa at 0 K to 34-65 GPa at 2000 K, suggesting high temperature is good for the stabilization of $F$$\overline{4}$3$m$ Sr₂B₅. Thus, $F$$\overline{4}$3$m$ Sr₂B₅ could be synthesized at moderate pressure by the large-volume press (LVP) since its stable range lies within the operating pressure and temperature range of LVP Ishii et al. (2019). Compared to high-$T_{\mathrm{c}}$ BCS hydride superconductors, which can only be realized by the diamond anvil cell with micrometer-sized samples, the production of $F$$\overline{4}$3$m$ Sr₂B₅ shows a certain advantage as LVP can provide much larger millimeter-sized samples for structure characterization and properties measurements. Moreover, we also found $F$$\overline{4}$3$m$ Sr₂B₅ is dynamically stable down to ambient pressure with anharmonic corrections (Fig. S11 SI ), implying it might be able to retain to 0 GPa with the proper quenching method. The $T_{\mathrm{c}}^{\mathrm{anh}}$ of $F$$\overline{4}$3$m$ Sr₂B₅ at 0 GPa is predicted to be even higher around 115 K (Fig. S12 SI ).

As part of efforts to develop high-temperature BCS superconductors, we designed a promising thermodynamically stable $F$$\overline{4}$3$m$ Sr₂B₅ with record-high $T_{\mathrm{c}}$ at moderate synthetic pressure ($\sim$ 40 GPa). We note this is the first fully metallic $sp^{3}$-hybridized boron framework example of metal borides, and we emphasize that the established B-B $\sigma$ bonding is a key to achieving high-$T_{\mathrm{c}}$ at low pressures. For instance, if B_D atom is also replaced by the strongly coupled $B_{4}$ unit in $F$$\overline{4}$3$m$ Sr₂B₅, the $T_{\mathrm{c}}$ might be even higher. Further investigation of other metal borides and lightweight compounds consisting of similar $\sigma$-bonding frameworks is expected which may lead to the discovery of potential high-$T_{\mathrm{c}}$ BCS superconductor (> 77 K) at ambient pressure.

In conclusion, our extensive first-principles structure searches for AE-metal (Mg, Ca, Sr, Ba) borides have revealed the appearance of stable Mg₃B₈, Ca₂B₅, Ca₃B₅, Sr₂B₅, and BaB below 100 GPa. Among them, $F$$\overline{4}$3$m$ Sr₂B₅ consisting of fully metallic $sp^{3}$-hybridized B-B $\sigma$ bonds exhibits potential high-$T_{\mathrm{c}}$ of 105 K at 40 GPa, with the possibility retaining to ambient pressure, that originates from the strong coupling between $\sigma$ electronic bands and rocking vibrations of $B_{4}$ units. This unique boron framework resembles the diamond structure, serving as a design guidance for future investigation. Our results provide a useful structural prototype for designing high-temperature superconductors at low pressures and will stimulate further experimental synthesis and theoretical predictions for a large variety of metallic $sp^{3}$-hybridized lightweight compounds.

This work is supported by by National Key Research and Development Program of China (Grant No. 2022YFA1402304), the National Natural Science Foundation of China (Grant no. 12022408, 12374008, 1202290458, 52288102, 52090024, 12074138, 12374007), the Interdisciplinary Integration and Innovation Project of JLU, Fundamental Research Funds for the Central Universities, Program for Jilin University Science and Technology Innovative Research Team (2021TD–05), Jilin Province Science and Technology Development Program (Grant No. YDZJ202102CXJD016), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB33000000) and computing facilities at the High-Performance Computing Centre of Jilin University.