Abstract
Atomically thin borophene has recently been synthesized experimentally, significantly enriching the boron chemistry and broadening the family of twodimensional (2D) materials. Recently, oxides of 2D materials have been widely investigated for nextgeneration electronic devices. Based on the firstprinciples calculations, we predict the existence of the superconductivity in honeycomb borophene oxide (B_{2}O), which possesses a high stability and could be potentially prepared by intrinsically incorporating oxygen into the recently synthesized borophene. The mechanical, electronic, phonon properties, as well as electron–phonon coupling of metallic B_{2}O monolayer, have been systematically scrutinized. Within the framework of the Bardeen–Cooper–Schrieffer theory framework, the B_{2}O monolayer exhibits an intrinsic superconducting feature with a superconducting transition temperature (T_{c}) of ~10.3 K, higher than many 2D borides (0.2–7.8 K). Further, strain can be utilized to tune the superconductivity with the optimal T_{c} of 14.7 K under a tensile strain of 1%. The superconducting trait mainly originates from the outofplane softmode vibrations of the system, which are significantly enhanced via the light O atoms’ incorporation compared to other 2D metalboride superconductors. This strategy would open a door to design 2D superconducting structures via the participation of light elements. We believe our findings greatly bloom the 2D superconducting family and pave the way for future nanoelectronics.
Introduction
Followed by theoretical prediction^{1}, atomically thin borophene analogs of twodimensional (2D) carbon materials, such as graphene, have recently been synthesized on Ag(111)^{2,3} and Au(111) substrates^{4}. Due to the unique mechanical and electronic properties, borophene can be utilized in various applications, such as electrode materials in rechargeable batteries^{5,6,7,8,9}, hydrogen storage^{10,11}, sensor^{12,13,14}, etc. These studies have significantly enriched the boron chemistry and broaden the family of 2D materials. It is known to be polymorphic metal monolayer with short covalent radius and sp^{2} hybridization, greatly different from its bulk phases with the semiconducting traits^{3,15}. Furthermore, borophene is demonstrated to be an intrinsic superconductor^{16,17,18}, showing an entirely different electronic property when compared with graphene. Next, intensive 2D superconductive borides, such as B_{2}C^{19}, Li_{2}B_{7}^{20}, tetrMo_{2}B_{2}, triMo_{2}B_{2}^{21}, tetrW_{2}B_{2}, hexW_{2}B_{2}^{22}, AlB_{6}^{23}, and XB_{6} (X=Ga, In)^{24}, provide heretofore the diversity in the context of motifs, electron–phonon coupling (EPC) and superconductivity. However, the T_{c}, especially in 2D metal borides, is low to be only 0.2–7.8 K. This significantly necessitate the exploration of 2D boronbased superconductors with enhanced superconductivity.
It should be noteworthy that the electrondeficient property of B atoms makes B–B bonds unstable in borophene monolayer, particularly under oxygenrich conditions^{25,26}. This calls for the investigation of the possible 2D “B_{x}O_{y}” materials when exposed to oxygen. By scrutinizing the oxygen adsorption and dissociation on freestanding borophene, Oadsorbed borophene exhibits an enhanced stability via the strong B–O interactions^{27}, in line with the characteristics of oxidation as reported in previous experiments^{2,3}. These phenomena are ultimately in favor of the formation of 2D boron oxides under ambient conditions. Besides, oxidation could help to tune the physical and chemical properties of 2D materials as well^{28,29,30}. Thus, borophene as electrondeficient monolayer is feasible to be embedded by oxygen via the formation of more stable B–O bonds^{31}. It is in this way which the 2D boron oxides could be obtained potentially. Inspired by this, more attentions have been focusing on these 2D boron oxides. Using the particleswarm optimization algorithm, zhang et al. have systematically explored the 2D boron oxide crystals, including compositions of B_{4}O, B_{5}O, B_{6}O, B_{7}O, B_{8}O, possessing attractive electronic features^{31}. Very recently, a 2D honeycomb boron oxide (B_{2}O) with planar monolayer is proposed and exhibits intriguing topological phase transition when subject to external strains^{32}. However, superconductivity in 2D boron oxides is yet to be reported experimentally and theoretically^{33,34,35,36}. Considering the intrinsic superconductivity in borophene, the successful exploration of superconductivity in 2D boron oxides would not only bloom the 2D boron family but also feasibly facilitate the experimental explorations for realistic applications.
Inspired by prior successes in exploring superconductivity in borophene^{16,17,18,37,38,39}, here we systematically study the superconductivity of 2D B_{2}O monolayer with the lowest formation energy^{31}. Using the firstprinciples calculations, the thermal and mechanical stability, electron structures, vibration modes, and superconductivity were discussed. The results show that B_{2}O is not only a metallic monolayer but also an intrinsic superconductor with superconducting transition temperature (T_{c}) of 10.3 K that is higher than those of the mostly reported 2D borides. Such superconductivity is attributed to the outofplane softmode vibrations of O atoms. Moreover, under the tensile strain of 1%, a large softened vibration mode appears along M–X direction, that, leads to an increase of T_{c} by 40%, showing the tunability of superconductivity in B_{2}O monolayer.
Results and discussion
The structure properties of B_{2}O
The artificial B_{2}O monolayer in this work was verified to be the global minimum structure by adopting particleswarm optimization^{40,41}. B_{2}O has a global minimum of energy of −6.80 eV/atom, ~0.48 and 0.50 eV smaller than the configurations with the 2nd and 3rd lowest energies, respectively (Supplementary Fig. 1a, b). However, the latter two structures are not dynamically stable, confirmed by the calculated phonon spectra (Supplementary Fig. 1c, d). Thus, we next only focus our attentions on the one with the global minimum energy. The title B_{2}O monolayer crystallizes in the orthorhombic lattice with space group Cmmm (No. 65), possessing a C_{2v} symmetry (Fig. 1a). The lattice parameters of B_{2}O are a = b = 3.93 Å. The B atom adopts a slightly distorted trigonal coordination with the bond angles of 107.8 and 126.1° as shown in Fig. 1b. The B–B bond distance is 1.71 Å, comparable with that of δ_{6} (1.62–1.87 Å)^{42}, χ_{3} (1.62–1.72 Å) and β_{12} borophene (1.65–1.75 Å)^{16}. The bond length of B–O is 1.34 Å, significantly shorter than that of B_{4}O (1.53 and 1.61 Å)^{31}, indicating the lower bond energy and higher bond strength. The B–B and B–O bonds exhibit the strong covalent bonding traits (Fig. 1c), further favoring the high stability within the B_{2}O plane.
The stability of B_{2}O crystal
The stability of 2D crystals is very important in predicted structures. Here, the cohesive energy (E_{coh}) and formation energy (E_{f}) are performed by
and
respectively, where \({E}_{{{\mathrm{B}}}_{{\mathrm{2}}}{\mathrm{O}}}\) is the total energy of primitive cell of B_{2}O, and E_{B} and E_{O} are the energies of isolated B and O atom, respectively. The μ_{B} is the chemical potential of χ_{3} borophene, and the μ_{O} is the chemical potential of O_{2}. The obtained E_{coh} and E_{f} are calculated to be 2.43 and −0.99 eV, respectively, indicating the exothermic process and experimental feasibility under suitable external conditions. Besides, the evolution of the free energy obtained from ab initio molecular dynamics (AIMD) simulations is exhibited in Supplementary Fig. 2. The average value of the free energy remains nearly constant with small fluctuations during the entire simulation period at about 1500 K (Supplementary Fig. 2a). After 10ps simulation, we found that there exists a sign of a structural disruption at about 1700 K (Supplementary Fig. 2b), producing a calculated melting temperature of 1500–1700 K. This melting temperature is higher than the previous report (1000 K)^{32}, suggesting a higher thermal stability and thus the potential applications in extreme hightemperature environment.
To further assess the mechanical stability of our structure, we calculated the elastic constants C_{ij} in the rectangle unit cell by
where E_{s} is strain energy, and the tensile strain is defined as \(\varepsilon =\frac{{a}{a}_{0}}{{a}_{0}}\), and a and a_{0} are the lattice constants of the strained and strainfree structures, respectively. ε_{xx} and ε_{yy} are the strains along the x and y directions, and ε_{xy} is the shear strain. The B_{2}O belonging to orthorhombic crystal has four independent elastic constants: C_{11}, C_{12}, C_{22}, and C_{66}, corresponding to the second partial derivative of strain energy with respect to the applied strain. In order to calculate the elastic stiffness constants, the E_{s} as a function of ε in the range of − 2% ≤ ε ≤ 2% were calculated. The results of the strain energy curves associated with uniaxial, biaxial, and shear strains are shown in Supplementary Fig. 3a. Then, the C_{ij} can be obtained with the aid of the VASPKIT^{43}, a postprocessing program for the VASP code. Our calculations estimate C_{11}, C_{12}, C_{22}, and C_{66} to be 42.57 N/m, 67.63 N/m, 237.59 N/m, and 11.58 N/m, respectively. Obviously, B_{2}O monolayer satisfies the Born criteria^{44}, C_{11} > 0, C_{66} > 0, and C_{11}C_{22} – C\({}_{12}^{2}>\) 0^{45}, which further confirms the mechanical stability of B_{2}O monolayer.
The inplane Young’s moduli (Y) along the x and y directions are obtained with the help of elastic constants as: Y\({}_{x}={\frac{{{\rm{C}}}_{11}{C}_{22}\,\,{C}_{12}^{2}}{{\rm{C}}}}_{22}=23.42\) N/m and Y\({}_{y}=\frac{{{\rm{C}}}_{11}{C}_{22}\,\,{C}_{12}^{2}}{{{\rm{C}}}_{11}}=130.40\) N/m. Apparently, the Young’s moduli are comparable with TiN (143 N/m)^{46}, MoS_{2} (123 N/m)^{47}, phosphorene (24–103 N/m) and silicene (62 N/m)^{48}. The Poisson’s ratio reflects the mechanical responses of the system against uniaxial strains and can be calculated as \({\nu }_{x}=\frac{{{\rm{C}}}_{12}}{{{\rm{C}}}_{22}}=0.29\) and \({\nu }_{y}=\frac{{{\rm{C}}}_{12}}{{{\rm{C}}}_{11}}=1.59\), indicating the large anisotropy in mechanical properties. To present a full understanding of the mechanical properties of B_{2}O monolayer, we calculated the orientationdependent Y and ν as a function of the polar angle θ (0–360°). For the orthogogonal 2D system, the strain parallel (ε_{∥}) and perpendicular (ε_{⊥}) to the θ direction induced by the unit stress σ(θ)(∣σ∣ = 1) can be expressed as^{49,50}
and
respectively, where \(\Delta ={{\rm{C}}}_{11}{{\rm{C}}}_{22}{{\rm{C}}}_{12}^{2}\), \(a=\cos\)(θ) and \(b=\sin\)(θ). Then, Y(θ) and ν(θ) are derived as
and
respectively. Clearly, both the variations of Y and ν show a spindlelike shape, indicating the fully anisotropic traits (Supplementary Fig. 3b, c). The Young’s modulus has a minimal value of 23.42 N/m along the x direction (θ = 0°), and a maximal value of 130.40 N/m along the y direction (θ = 90°). Meanwhile, the ν(θ) of B_{2}O increases monotonically from 0.29 (θ = 0°) to 1.59 (θ = 90°). These anisotropies are associated with the bond interactions of B–B and B–O bonds in the two directions.
Electronic properties of B_{2}O monolayer
In order to probe the nature of charge transfer of B_{2}O monolayer, we also calculated the difference charge density Δρ, as shown in Fig. 1d. The charge transfer mainly occurs from less electronegative B to more electronegative O atoms. Although the σ states of O atoms gain electrons, whereas the π states partially loss electrons as well. There are significantly delocalized charge accumulations within B–O bond, indicating the covalent bonding feature. According to the Bader analysis^{51}, the net charger transfer from B to O is 0.79 electron per atom, in consistence with above distribution of difference charge density. The ionic feature of B_{2}O can be thus represented as B_{2}^{0.79+}O^{1.58−}, showing charge transfer predominantly from B to O atoms via B–O bonding interactions.
The B_{2}O monolayer is metallic with two bands crossing the Fermi level (Fig. 2a), which is supported by the Fermi surface distributions along the high paths (Fig. 2b). From the orbitalresolved band structure, the bands around the Fermi level are mainly composed of the Bp and Op orbitals. In Supplementary Fig. 4, we can clearly see that the orbital hybridization near the Fermi level mainly stems from Bp_{x,y} and Bp_{z} states, followed by some contributions from Op_{x,y}, Op_{z}, and Bs states. Thus, the metallic nature of B_{2}O monolayer is essentially dominated by the Bp orbitals. In addition, two dirac cones (DCs) exist around the Fermi level. The orbitalresolved band structures of the B_{2}O monolayer are presented in Supplementary Fig. 5 to understand the origin of DPs. The DC1 mainly consists of Bp_{y} and Op_{z}, and the DC2 involves the dominant contributions from Bp_{z} and Op_{y}. Nevertheless, the s, p_{x} orbitals of two atoms are not responsible for the formation of the DCs (Fig. 2a; Supplementary Fig. 5). The effects of spinorbit coupling and Heyd–Scuseria–Ernzerhof (HSE06)^{52} are evaluated to play a negligible role in the formation of DCs^{32}. Even, the DCs are still well maintained with strain reaching up to 3% (Supplementary Fig. 8b), indicating that the 2D B_{2}O is a robust Dirac material. To assess the electronic transport properties of B_{2}O monolayer, we further calculated the Fermi velocity (V_{F}) within PBE level along Y–M by using the equation V_{F} = \(\frac{\partial E}{\hslash \partial k}\), where the \(\frac{\partial E}{\partial k}\) is the slop of the linear band structure and the \(\hbar\) is the reduced Planck’s constant. The calculated V_{F} closing to DC2 are 9.6 × 10^{5} and 6.8 × 10^{5} m/s, respectively, larger than and comparable with graphene (8.22 × 10^{5} m/s)^{53}. This high value of V_{F} suggests B_{2}O monolayer to be possess a ultrahigh carrier mobility and would facilitate the future electronics. To probe the effects of the percentage of Hartree–Fock (HF) functionals on electronic properties, screened exchange hybrid density functional of HSE06 were carried out as well (Supplementary Fig. 6). Upon increasing the fraction of HF in the HSE06 calculations, the dispersion of band structure shows a small variation when compared with the PBE results, and the main band traits, such as the two DCs, are maintained. This result suggests that HF plays a negligible role in the electronic structure of system. So, we only calculated and analyzed the superconductivity next on the PBE level.
Phonon and superconductivity of B_{2}O crystal
The phonon spectrum of B_{2}O monolayer shown in Fig. 3 reveals no imaginary phonon modes, indicating that the rhombic phase is kinetically stable, which is in line with previous result^{32}. The phonon k ⋅ p theorem is used to sort the phonon branches based on the continuity of the eigenvectors of vibration modes^{54,55,56},
where \({e}_{k,\sigma }^{* }(j)\) is the displacement of atom j in the eigenvector of (k, σ) vibration mode, and Δ is a small wave vector. As indicated in Fig. 3a, the outofplane (ZA), inplane transverse (TA), and inplane longitudinal (LA) modes constitute the three acoustic branches for B_{2}O.
According to the decomposition of the phonon spectrum with respect to the vibration directions of B and O atoms (Fig. 3a), three acoustic branches dominate the lowfrequency region (below 400 cm^{−1}), where the main contributions are from inplane and outofplan of modes of O atoms. Meanwhile, the two lowest optical branches consisting of the outofplan of modes of Bz also make large contributions to this range. Generally, due to the subtle differences in atomic weight, the outofplane modes of Bz and Oz mostly occupy the lowfrequency region. The midfrequency region from 400 to 900 cm^{−1} are related entirely to the inplane of modes of Bxy and the Bz. Moreover, the vibration frequencies larger than 900 cm^{−1} originate from the outofplan of modes of Bxy and Oxy. The highest vibration frequency, 1471 cm^{−1}, is much larger than that of Mo_{2}B_{2} (880 cm^{−1})^{21}, W_{2}B_{2} (920 cm^{−1})^{22}, Li_{2}B_{7} (1120 cm^{−1})^{20}, AlB_{6} (1150 cm^{−1})^{23}, β_{12} borophene (1200 cm^{−1})^{16,57}, χ_{3} borophene (1290 cm^{−1})^{16,58}, and B_{2}C (1365 cm^{−1})^{19}, it is even comparable with that of δ_{6} borophene (1411 cm^{−1})^{42}. Such a high frequency is consistent with the strong covalent bonding, suggested by the former results of ELF (Fig. 1c) and projected phonon density of states (PhDOS) (Fig. 4a).
The results of the PhDOS, Eliashberg spectral function α^{2}F(ω), the EPC constant λ, T_{c}, and the derivatives of T_{c} are presented in Fig. 4. The Eliashberg spectral function exhibits that four major peaks are located at 88.3, 119.3, 210.2, and 288.2 cm^{−1}, respectively, in the lowfrequency region. As shown in Fig. 3b, the lowfrequency mode phonons contribute mainly to the EPC, accounting for 78.5% of the total EPC (λ = 0.75). The first peak of α^{2}F(ω) is mainly responsible for this part, ~51.9% of the total EPC. In the frequency range of 83–120 cm^{−1}, the large values of λ_{qν} along the M–X − Γ directions are visible and lead to the first two largest peaks of the α^{2}F(ω) (Figs. 3b and 4b). As a consequence, λ(ω) increases rapidly in this range. In order to probe the underlying causes, we analyze the vibration modes with the largest value of λ_{qν} in this range (see Fig. 3b). Clearly, the outofplane vibrations of B and O atoms contribute to the increase of λ_{qν}. Besides, in midfrequency region (400–800 cm^{−1}), the phonons contribute the rest of the EPC by ~21.5%. However, the contributions of highfrequency phonons are negligible, which is similar to Mo_{2}B_{2}^{21}, W_{2}B_{2}^{22}, AlB_{6}^{23}, β_{12} borophene^{16,57}, and χ_{3} borophene^{16,58}. Using the McMillian–Allen–Dynes formula^{59}, the frequencydependent superconducting transition temperature T_{c}(ω) is obtained from Fig. 4c. Its derivative also exhibits the four main peaks in low frequency, which consistent with distributions of α^{2}F(ω). The B_{2}O monolayer is a mediumcoupling superconductor with λ of 0.75 according to the role proposed by Allen et al^{59}. and possesses a T_{c} of 10.35 K, which is higher than those of LiC_{6} (5.9 K)^{60,61}, C_{6}CaC_{6} (4.0 K)^{62,63}, and CuBHT (3.0 K)^{64} that their values of T_{c} were determined in experiments. Moreover, the T_{c} of B_{2}O is also higher than recently reported 2D boride superconductors using a typical value of μ^{*} = 0.1, such as Li_{2}B_{7} (6.2 K)^{20}, B allotrope (6.7 K)^{17}, tetrMo_{2}B_{2} (3.9 K), triMo_{2}B_{2} (0.2 K)^{21}, tetrW_{2}B_{2} (7.8 K), hexW_{2}B_{2} (1.5 K)^{22}, rectGaB_{6} (1.7 K), rectInB_{6} (7.8 K), hexInB_{6} (4.8 K)^{24}, and AlB_{6} (0.95 K)^{23}. This increase of T_{c} can be rationalized by the fact that the vibrations of B atoms are significantly enhanced via incorporating light O atoms into the monolayer, in great contrast to the constraining effect associated with heavier metal atoms within 2D metalboride superconductors. This provides clues for us to design 2D superconducting systems with light elements and opens the road toward further improvement of 2D superconducting feature. It is true that some intrinsic 2D stable B structures such as δ_{6} (27.0 K), χ_{3} (24.7 K), and β_{12} borophene (18.7 K)^{16} show superconductivity with higher T_{c}. This is not in contrast to our designing strategy of 2D superconductors via the participation of light elements: The pure 2D boron structures could be regarded the extreme phase of borides by introducing a more light "B" element into 2D sheet in the form of "B_{x}B" when compared with the O’s incorporation in B_{2}O monolayer. Here, the T_{c} of borophene is also calculated with μ^{*} = 0.1. To explore the effect of the μ^{*} on the T_{c} of B_{2}O monolayer, we calculate the T_{c} by varing μ^{*} from 0.08 to 0.15 (Supplementary Fig. 7). As expected, T_{c} would decrease monotonically decreasing form 11.9 K to 6.8 K upon the increase of μ^{*}.
The oxidation process of black phosphorene could be wellcontrolled by the assistance of Laser^{65} and borophene has been successfully synthesized on the substrates^{3,4}. Thus, the B_{2}O monolayer may be obtained on borophene substrate by oxidation^{32}. To simulate the real samples grown on substrates with different lattice constants, we applied the inplane biaxial strain to the B_{2}O monolayer. The biaxial strain can be calculated by \(\xi =\frac{{a}{a}_{0}}{{a}_{0}}\times 100 \%\) (positive value means tensile strain, while negative one indicates compressive strain). In our calculations, the lattice constants are changed from −1% to 3%, and atomic coordinates are fully relaxed in each case. The band structures and phonon spectra are plotted in Supplementary Fig. 8. By confirmed the phonon spectra in Supplementary Fig. 8a, B_{2}O is stable under the tensile strain from 1% to 3%. As indicated in Supplementary Fig. 8b, the tensile strain has few influence on the band structures around the Fermi level, the same as the N(E_{F}) (Fig. 5b; Supplementary Fig. 8b). While, the phonon spectra shift to some extent under applied tensile strain. Especially, compared with freestanding sample, appearing a large soften mode along M–X at strain of 1%, is a good phenomena improving the superconductivity. The variations of superconductive parameters [N(E_{F}), ω_{log}, λ, and T_{c}] under series of strain are exhibited in Fig. 5. Along with the increasing tensile strain, the ω_{log} first decrease until strain greater than 1%, and then it goes up conversely. While the λ and T_{c} vary in an inverse way, and the N(E_{F}) varies a little. When tensile strain equals 1%, the λ and T_{c} can be increased to be 1.04 and 14.7 K, respectively, reaching the maximum values in this strain region. However, the bigger tensile strain suppresses the superconductivity. So, strain engineering offers an effective way to tune the superconductivity and facilitates the potential application in future nanodevices.
In summary, within the framework of the densityfunctional theory (DFT) and Bardeen–Cooper–Schrieffer (BCS) theory, we have systematically investigated the mechanical and electronic properties, phonon vibrations as well as superconductivity of the proposed 2D honeycomb borophene oxide, B_{2}O. The B_{2}O monolayer exhibits a high stability and possesses two Dirac cones near the Fermi level, and is an intrinsic BCStype superconductor with a T_{c} of ~10.3 K. This T_{c} is higher than mostly reported boride superconductors. The superconducting trait is attributed to the outofplane vibration modes of B and O atoms. Upon applying a tensile inplane strain of 1%, the T_{c} can achieve the maximum value of 14.7 K, which is associated with the large soften mode appearing along M–X direction in ZA branch. Thus, these interesting results would further trigger efforts on 2D superconducting materials.
Methods
Firstprinciples calculations
The firstprinciples calculations based on the DFT were performed through the Vienna ab initio simulation package (VASP)^{66,67} and the QuantumESPRESSO code. After the full convergence tests, the exchange correlation interaction was simulated within the generalized gradient approximation (GGA)^{68,69} with the Perdew–Burke–Ernzerhof (PBE)^{70}type pseudopotential. The electronic wave functions were expanded via the plane wave basis set with a energy cutoff of 600 eV. The Γcentered 15 × 15 × 1 kpoint mesh were adopted using the Monkhorst–Pack method. To avoid the interaction between adjacent monolayers, the vacuum thickness was set to be 15 Å along the z direction. The structure were fully relaxed until the total energy and force on per atom were <10^{−5} eV and 0.01 eV/Å, respectively^{71,72}. The EPC and superconductivity were calculated by the QE within the densityfunctional perturbation theory (DFPT)^{73} and the BCS theory^{74}. The optimized normconserving Vanderbilt pseudopotentials^{75} were used to model the electronion interactions. The kinetic energy cutoff and the charge density cutoff of the plane wave basis were chosen to be 80 and 320 Ry, respectively. Selfconsistent electron density was evaluated by employing 24 × 24 × 1 k mesh with a Methfessel–Paxton smearing width of 0.02 Ry. The phonon calculations were carried out on the 6 × 6 × 1 q mesh. Meanwhile, the convergences of λ and T_{c} are tested and shown in Supplementary Fig. 9, verifying that the q mesh of 6 × 6 × 1 is large enough to used in our calculations.
Structure screening and AIMD calculations
The particleswarm optimization (PSO) scheme, as implemented in the CALYPSO code^{40,41}, was adopted to search for the global minimum structure for the B_{2}O compound. In the PSO calculations, both planar and buckled structures of B_{2}O were considered, and the population size and the number of generations were set to be 30. Through the highthroughput calculations, thousands of different structures of B_{2}O were generated and ranked by CALYPSO code in order of enthalpy from low to high. The electronic structure calculations were performed through the VASP. We also performed the AIMD simulations at a series of temperatures (300, 500, 700, 900, 1100, 1300, 1500, and 1700 K) with constant number, volume, temperature (NVT) ensemble, lasting for 10 ps with a time step of 1 fs to assess the thermal stabilities of B_{2}O monolayer. The 3 × 3 × 1 supercell was adopted and the temperature was controlled using the Nosé–Hoover thermostat^{76}.
Superconductivity calculations
Within the BCS and Migdal–Eliashberg theories^{77,78} framework, to examine the contribution to λ from individual phonon modes, the magnitude of the EPC λ_{qν} can be calculated by
where γ_{qν}, ω_{qν}, and N(E_{F}) are the phonon linewidths, the frequency of a lattice vibration with crystal momentum q in the branch ν and the density of states (DOS) at the Fermi level, respectively^{79}. In addition, the phonon linewidths γ_{qν} can be estimated by
where Ω_{BZ} is the volume of Brillouin zone (BZ); ϵ_{kn} and ϵ_{k+qm} are the Kohn–Sham energy, ϵ_{F} is the Fermi energy, and \({{\rm{g}}}_{{\boldsymbol{k}}n,{\boldsymbol{k}}+{\boldsymbol{q}}m}^{\nu }\) denotes the EPC matrix element. Moreover, according to the linear response theory^{59} the \({{\rm{g}}}_{{\boldsymbol{k}}n,{\boldsymbol{k}}+{\boldsymbol{q}}m}^{\nu }\) can be determined selfconsistently. Subsequently, the Eliashberg electron–phonon spectral function α^{2}F(ω) and the cumulative frequencydependent EPC function λ(ω) can be calculated by
and
respectively.
The logarithmic average frequency \({\omega }_{\mathrm{log}\,}\) and the superconducting transition temperature T_{c} can be calculated as follows:
and
where μ* = 0.1, a typical value of the effective screened Coulomb repulsion constant^{16,57,80,81,82,83,84}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
L.Z. acknowledges the financial support from the University of Electronic Science and Technology of China. P.F.L. and B.T.W. acknowledge the PhD Startup Fund of Natural Science Foundation of Guangdong Province of China (Grant No. 2018A0303100013). L.Y. thanks Y.K.L. from Chinese University of Hong Kong for some discussions.
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L.Y. performed the conceptualization, data curation, and writing—original draft and validation. P.F.L. performed investigation and validation. H.L. performed the formal analysis. Y.T. made software and investigation. J.H. did investigation. B.W. performed the supervision and validation. L.Z. performed the conceptualization, writing–reviewing, and editing. All authors discussed and analyzed the results and commented on the paper.
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Yan, L., Liu, PF., Li, H. et al. Theoretical dissection of superconductivity in twodimensional honeycomb borophene oxide B_{2}O crystal with a high stability. npj Comput Mater 6, 94 (2020). https://doi.org/10.1038/s41524020003659
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