Prediction of Superconducting RbBSi  Compounds under Pressure

The development of superconductors at high temperatures or even ambient temperature has been a long-standing goal in the scientific community since 1911^[1]. Superconductors have great applications in various fields, such as power transmission, magnetic resonance imaging, and quantum computing^[2–4]. In recent years, there has been a major breakthrough in the field of new phonon-mediated superconductor, specifically in the study of clathrate hydrides. Clathrate hydrides attract attention due to their unique crystal structure and high superconducting critical temperatures above 200 K under high pressure^[5–8]. One noteworthy discovery is CaH₆, and this hydride exhibits a superconducting critical temperature (T_c) of 215 K at 172 GPa^[9–10]. This is followed by the discovery of LaH₁₀, and it has T_c of 260 K at about 170 GPa^[11–13]. While these high-T_c hydrides show promising superconducting properties, they require comparable high pressures to synthesize and are typically unrecoverable at ambient pressure, limiting their practical applications. Researchers continue to explore novel materials and mechanisms that can exhibit superconductivity at more accessible conditions. Such advancements would greatly enhance the feasibility and utilization of superconductors in various technological applications.

Encouragingly, some non-hydrogen clathrates, in particular the sp³ bonded ones with light elements, have been suggested to be superconducting at moderate or even ambient pressure. Many doped albite structures, such as XC₆^[14–15], X(BN)₆^[16], XB₃C₃^[17], and XB₃Si₃^[18], where X represents different doping elements, as well as FC₃₄^[19], has been predicted to be good superconductors at ambient pressure. For example, T_c calculated for NaC₆ is 116–127 K at ambient pressure. It is noteworthy that with the guiding from particle-swarm structure prediction method, Zhu et al.^[17] successfully synthesized a thermodynamically stable carbon-boron sp³ bonded clathrate at high pressures near 50 GPa, namely SrB₃C₃. The calculations further suggested that SrB₃C₃ is a superconductor, with the superconducting transition temperature of 40 K^[20–22]. Carbon and silicon are elements of group-14 in the periodic table of elements^[23]. They have four valence electrons, allowing them to form covalent bonds with other elements by sharing electrons. Moreover, they have certain similarities in atomic structure and chemical properties^[24–25]. In the silicon analogues, RbB₃Si₃ is theoretically proposed to be thermodynamically stable between 7 and 35 GPa, and to be metastable at 1 atmospheric pressure, with T_c estimated to be 14 K^[18]. Therefore, it is of great interest to explore whether the RbBSi compounds have the potential for superconductivity under ambient pressures.

In this work, we employed CALYPSO (crystal structure analysis by particle swarm optimization) method to predict the stable structure of RbBSi within the pressure range from zero to 100 GPa, and studied their physical properties by first-principles calculations.

1. Methods

We explored the properties of RbBSi in the pressure range of 0−100 GPa using CALYPSO code^[26–29], which has been developed specifically to explore the potential energy surfaces of chemical systems and find the global minima. The structural relaxations and electronic properties were calculated using density functional theory with the Perdew-Burke-Ernzerhof generalized gradient approximation as implemented in the VASP code^[30–31]. We used projector augmented wave (PAW) potentials^[32] in which the valence electrons for the Rb, B and Si atoms were represented as 4s²4p⁶5s¹, 2s²2p¹, and 3s²3p². In order to ensure the enthalpy convergence, the cutoff energy was adopted 500 eV. The Monkhorst-Pack k-point meshes are 8×8×2 for P6₃/mmc-RbBSi, 3×8×2 for C2/m-RbBSi, and 8×8×3 for P4/nmm-RbBSi, respectively. The phonon dispersion curves were used to verify the dynamic stability of RbBSi via a supercell approach as implemented in PHONOPY package^[33]. The electron-phonon coupling constant was obtained from first-principles density-functional perturbation theory as implemented in QUANTUM ESPRESSO package^[34].

2. Results and Discussion

The RbBSi crystal structure prediction was successfully conducted in the pressure range of 0－100 GPa, and three phases (P6₃/mmc, C2/m and P4/nmm) were discovered. As shown in Fig. 1(a), P6₃/mmc-RbBSi at standard atmospheric pressure has a layered hexagonal structure that consists of alternate B-Si honeycomb layers separated by a layer of interstitial Rb atoms. The bond length between boron and silicon is 2.049 Å, forming a stable hexagonal structure. In the pressure range of 17 GPa to 33 GPa, the C2/m phase was found to be more stable (see Fig. 1(b)). In the C2/m-RbBSi structure, there are six membered rings and four membered rings composed of folded B-Si layers. The layers in these structures are separated by Rb atoms in the gap between the two layers. The lengthes of the boron-silicon bond in the four-membered ring and in the six-membered ring are 2.095 and 2.115 Å, respectively, and the length of the boron-boron bond is 1.678 Å. As the pressure increases, the P4/nmm-RbBSi phase becomes more stable above 33 GPa (see Fig. 1(c)). Different from C2/m-RbBSi, the space group of P4/nmm-RbBSi changes from monoclinic to tetragonal, and the folded B-Si layer for P4/nmm-RbBSi is only composed of four-membered rings. The bond length of the boron-silicon bond is 2.098 Å. The detailed lattice parameters and atomic positions of the structures are given in Table 1.

Figure  1.  The predicted crystal structures of the RbBSi: (a) P6₃/mmc, (b) C2/m, and (c) P4/nmm

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Table  1.  Detailed structure information of the predicted RbBSi compounds at standard atmospheric pressure

Space group	Lattice parameters	Atomic positions
P63/mmc	a = b = 3.549 Å，c = 11.365 Å α = β = 90°，γ = 120°	Rb 2a (0, 0, 0) B 2d (0.667, 0.333, 0.250) Si 2c (0.667, 0.333, 0.750)
C2/m	a = 9.004 Å，b = 3.518 Å，c = 11.226 Å α = γ = 90°，β = 109.3642°	Rb 4i (0.532, 0, 0.770) B 4i (0.903, 0, 0.489) Si 4i (0.770, 0.500, 0.424)
P4/nmm	a = b = 3.752 Å，c = 11.170 Å α = β = γ = 90°	Rb 8j (0, 0.500, 0.736) B 8j (0, 0.500, 0.084) Si 4d (0.500, 0.500, 0)

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We also calculated the formation enthalpy (see Fig. 2), and systematically studied the stability of the three phases of RbBSi compounds. The formation enthalpy for RbBSi were defined as: ΔH = H(RbBSi) − H(Rb) − H(B) − H(Si). For Rb, we used the enthalpy from bcc → fcc → I4₁/amd → Cmca structure transitions, with the phase transitions occurring at 8.2, 15.1 and 41.0 GPa, respectively^[35]. For B, we used the enthalpy from $R\overline 3 m$ → Pnnm (γ-B) → Cmca (α-Ga) structure transitions, with the phase transitions occurring at 17.7 and 91.3 GPa, respectively^[36]. For Si, we used the enthalpies from $Fd\overline 3m$ (dia-Si) → Imma → P6₃/mmc (hcp) → $Fm\overline 3m$ structure transitions, with the phase transitions occurring at 8.9, 40.0 and 83.9 GPa, respectively^[37]. It can be seen from Fig. 2 that as the pressure increases, the crystal structure phase changes from the P6₃/mmc phase to the C2/m phase at 17 GPa . Once the pressure reaches 33 GPa, the crystal structure phase with the lowest energy is the P4/nmm phase.

Figure  2.  Calculated formation enthalpies of RbBSi compounds with the P6₃mmc, C2/m, and P4/nmm structures

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The phonon calculation is a reliable measure to confirm the dynamic stability of studied materials. Therefore, we calculated phonon dispersion curves for P6₃/mmc at 0 and 10 GPa, for C2/m at 0 and 20 GPa, and for P4/nmm at 0 and 40 GPa, as shown in Fig. 3. Our phonon calculations showed that all the three phases are dynamically stable because of the absence of any imaginary frequency vibrations in the whole Brillouin zone.

Figure  3.  Phonon dispersion curves of the three structre of RbBSi under different pressures: (a) P6₃/mmc at 0 GPa, (b) C2/m at 0 GPa, (c) P4/nmm at 0 GPa, (d) P6₃/mmc at 10 GPa, (e) C2/m at 20 GPa, (f) P4/nmm at 40 GPa

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We next explored the electronic properties of the predicted RbBSi compounds via electronic band structures and densities of electronic states (DOS). As shown in Fig. 4, it can be observed that there is a significant consistency between the band structures and DOS. As shown in Fig. 4(a)−Fig.4(c), the bands cross over the Fermi level, indicating that these phases are metallic, which agrees well with the results of DOS (see Fig. 4(d)−Fig. 4(f)). In addition, the major contribution of metallicity comes from the B-p state and the Si-p state.

Figure  4.  Calculated band and densities of electronic states of RbBSi: (a) and (d) P6₃/mmc at 10 GPa, (b) and (e) C2/m at 20 GPa, (c) and (f) P4/nmm at 40 GPa

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Turning to the phonon and electron-phonon coupling in P4/nmm-RbBSi, the calculated phonon dispersion results are shown in Fig. 5. No imaginary phonon modes are present in the entire Brillouin zone, indicating the dynamic stability of this crystal structure. We then computed the Eliashberg spectral function α²F(ω), where α represents the electron-phonon interaction strength and ω represents the frequency. The overall isotropic electron-phonon coupling parameter λ can be obtained via a simple integration in the frequency domain. λ is 0.84, which is quite large for phonon-mediated superconductors. These calculations suggest that P4/nmm-RbBSi is a superconductor with T_c of 14.4 K at standard atmospheric pressures. The superconducting transition temperature is reduced to 0 K at 40 GPa.

Figure  5.  Calculated phonon dispersions, projected phonon density of states and Eliashberg spectral function α²F(ω) of the P4/nmm-RbBSi at standard atmospheric pressure

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In addition, it was found that P6₃/mmc-RbBSi also exhibits superconducting properties, and the superconducting transition temperature under standard atmospheric pressure is 5.2 K. Its λ is 0.50. However, its superconducting transition temperature is reduced to 3.3 K at 10 GPa, and the corresponding λ is 0.44.

3. Conclusions

In summary, we have extensively explored the RbBSi within the pressure range of 0−100 GPa using CALYPSO method. It is found that RbBSi has three thermodynamically stable phases, denoted as P6₃/mmc-, C2/m-, P4/nmm-RbBSi. All the phases studied are dynamically stable in the pressure ranges of 0−10 GPa, 0−20 GPa and 0−40 GPa, respectively. Based on our calculations of formation enthalpy and density of electronic states, it is shown that these structures exhibit metallic properties. Electron-phonon calculations predicted that P4/nmm-RbBSi would be a superconductor with T_c of 14.4 K at standard atmospheric pressures. Our current findings predict a superconductor with synthetic potential, which provides a driving force for further research and development of superconducting materials.