Electron–Phonon Superconductivity in Boron‐Based Chalcogenide (X= S, Se) Monolayers

Electron–phonon mediated superconductivity is deeply investigated in two boron based monolayer materials, namely, B3S$B_{3}S$ , a metal exhibiting the ability to superconduct, and a new metal, B3Se$B_{3}Se$ , presenting perfect kinetic stability. Calculations based on density functional perturbation theory combined with the maximally localized Wannier function also reveal that both materials exhibit anisotropic planar hexagonal structure like graphene. The key parameters involved in the superconductor behavior are all calculated. The electronic density in the Fermi surface is given to provide the environment for enhanced electron–phonon coupling. The longitudinal and transverse vibration modes of optical phonons mainly contribute to the electron–phonon coupling strength. Furthermore, the binding energy between the bosonic Cooper pair superfluid is quantified and determined. The critical temperature for the two materials is 20 and 10.5 K, respectively. The results obtained show the potential use of such materials for superconducting applications.


Introduction
Since the discovery of graphene, [1,2] bi-dimensional materials have aroused great interest. Group-IV monoatomic sheets, including silicene, [3] germanene, [4] and stanene, [5] were all fabricated using several sophisticated methods such as chemical vapor deposition (CVD), ultrahigh vacuum by MBE-like methods, and molecular beam epitaxy. Silicene and germanene probes in therapeutic applications in nanomedicine among others [6] while stanene shows potential use for spintronic and electronic devices. [7] Monolayers based on group-III-elements, like borophene [8] and aluminene, [9] can exist in different configurations and exhibit a metallic character. Their synthesis on substrates reveals their potential applications in sensing, terahertz shielding, optoelectronic devices, and high-efficiency energy storage technologies. [10] Other monoelement materials promise wide implementation, namely, in spintronics for arsenene, [11] in chemical or bio-sensing for phosphorene, [12] in surface plasmon resonance sensor for antimonene [13] among others.
Recently, graphene-based materials have enriched the list of 2D hexagonal lattices for their use in optoelectronics, catalysis, carriers transport, and batteries. Ge-and Si-binary compounds (GeC and SiC) are semiconductors with strong excitonic effects and high binding energy. [14,15] C 2 N monolayer is used as gas sensing for NH 3 and NO. [16] BC 3 is considered for metal-free photocatalysts with a visible light absorption. [17] Ultrahigh carrier mobility is detected in M 2 C 3 monolayers with (M = As, Sb, and Bi), [18] while vanadium carbide V 2 C monolayer is predicted as anode material for high-performance lithium-ion batteries. [19] The topological property of 2D hexagonal OsC at room temperature makes it suitable for high-speed transport devices. [20] Driven by the mechanical or liquid exfoliation technique chemical vapor deposition (CVD) and hydrothermal synthesis, joint efforts have been made in order to develop the large-scale growth of 2D materials and their novel devices. In 2017, the Janus transition metal chalcogenides MoSSe mono-layer was first prepared then successfully realized. [21,22] Group IIIA hexagonal monochalcogenides (AX) (with A = Ga, In, Tl, and X = S, Se, Te) gained significant interest after the experimental synthesis of GaS, GaSe, and InSe 2D materials. [23] The strong visible-light www.advancedsciencenews.com www.ann-phys.org absorbance of Janus group-III chalcogenide monolayers makes them potential for water-splitting photocatalysts. [24] Dichalcogenide transition metal MoS 2 monolayer has been grown on SiO 2 substrates [25] and integrated in spintronics devices via Rashba effect. [26] Ultrafast laser spectroscopy revealed a considerable photo carrier dynamics in MoS 2 compared to other dichalcogenide materials like WS 2 monolayer. [27] In addition, 2D boron chalcogenide (BX) materials, containing at least one chalcogen atom per unit cell, have attracted much interest for their superior thermoelectric efficiency. BS and BSe contain two phases, 1T and 2H, which exhibit a very high buckling parameter varying in the range [3.4-4.2]Å and indirect large band gap. [28] The 2H configuration is similar to the h-BN lattice, while in the 1T configuration the chalcogen atoms are placed in the center of the hexagons. The B 2 S compound, in a 1:2 ratio, includes two sulfur atoms and four boron atoms in a hexagonal ring, leading to a slightly distorted honeycomb structure and an orthogonal primitive cell. [29] Like planar graphene, this B 2 S material exhibits Dirac cone and robust wettability toward electrolytes. [30] The planar hexagonal structure with intrinsic metallicity of the B 3 S monolayer can be maintained after lithium adsorption, indicating a high storage capacity that makes the B 3 S compound an ideal candidate for a high-performance anode material in lithium-ion batteries. [31] However, Li-decorated B 3 S are high capacity materials for hydrogen storage. [32] Multiple physical phenomena, such as the superconductivity, electrical resistivity, carriers and heat transport, and Kohn effect, are dominated by the electron-phonon (e-ph) coupling in 2D materials. [33,34] Unlike graphene having Kohn anomalies, [35] silicene and germanene exhibit negligible EP interaction [36] while stanene presents strongly dependent EPC on the electron wave vector, which is interesting for transport phenomena. [37] The electron-phonon process gives rise to a high scattering rate in silicene and hot carrier thermalization in 2D carbon based-hybrids XC (with X=Si, Ge, Sn). [38] Furthermore, the strong (e-ph) coupling strength detected in hole-doped InSe monolayer gives rise to unconventional temperature-dependent optical excitation, [39] while it significantly reduces the lattice thermal conductivity for transitional metal dichalcogenide MoS 2 and PtSSe sheets. [40] The creation of Cooper pairs gives rise to superconductivity in some class of nanomaterials. One can distinguish two main kinds of superconductors, namely the conventional lowtemperature superconductors and the unconventional hightemperature ones, such cuprate superconducting materials. [41] Superconducting magnetic energy storage (SMES) has been developed as a new generation for use in high-power devices. [42,43] In addition, combining semiconductors MoS 2 to superconductors monolayer opened up a path to monolayer semiconductors as a platform for superconducting hybrid devices. [44] Interestingly, superconductivity has been induced into 2D materials since graphene has been tuned into superconductor with a critical temperature ranking from T c = 5.1 to T c = 7.6 K by lithium decoration or by stacking layers, respectively. [45] Moreover, it has been reported in ref. [46] that aluminene can superconduct with temperatures below 8.8, and 10 K for strained silicene. [47] Due to its metallic character, borophene in a rectangular buckled configuration presents a high anisotropy with a critical temperature up to 22 K. [48] This breakthrough spearheaded the discovery of superconductivity in many binary compounds such as MgB 2 with the deposition of a few layers, which get 30 K [49] and hexagonal MB 6 (M = Mg, Ca, Sc, Ti). [50] Maximally localized Wannier functions MWLFs within density functional perturbation theory are used to compute properties related to the electron-phonon coupling on very fine electron (k) and phonon (q) wave vector grids in order to enhance the accuracy of calculations by using a supercell with a localized basis set. The electron and phonon self energies are extracted and also the phonon linewidths are plotted, using the Migdal-Eliashberg formalism for low-temperature superconductors. The spectral function and the electron-phonon coupling strength are calculated and are found to give accurate analysis of the electron scattering with the vibrational modes of phonons. We also investigate the critical temperature expressed by the Mc Millan formula at a fixed screened Coulomb repulsion and also the superconducting gap mentioned as the energy of bonding for the Cooper pair which quantifies the degree of superconductivity. Furthermore, we perform a comparative analysis for both materials.
This work is organized as follows: in Section 3, the computational method is given; then we move to the discussion of the obtained results, and we compare them with other materials. We finally end with a conclusion.

Computational Method
The calculations are based on density functional theory (DFT) implemented in Quantum Espresso code. [51,52] The projector augmented wave (PAW) [53] within generalized gradient approximation GGA [54] are used for the exchange-correlation of electrons. The kinetic energy and charge density cutoff are set to be 60 and 720 Ry, respectively. For both boron-based chalcogenide structures, the self-consistent calculation uses a k-point grid of a Monkhorst pack K-mesh of 10 × 10 × 1 to sample the Brillouin zone (BZ) for integrations in reciprocal space. In order to avoid the interlayer interaction, a vacuum space of 28 Å along the z axis is considered. The phonon dispersions and dynamical matrix are calculated within the density functional perturbation theory with 20 × 20 × 1 q-points meshes. Thermal stability is also examined by calculating the formation energy and employing molecular dynamics (MD) simulations using the Nosé Hoover method [55] with a time step of 2 fs for 5 × 5 × 1 supercells constructed from the optimized geometries of primitive cells.
For the electron-phonon coupling, the EPW code [56,57] is employed using Wannier functions. For the superconducting physical quantities, we increase the grid of K-mesh points to 70 × 70 × 1 and of q-points to 35 × 35 × 1 to get more accurate results.
Recall in passing that the calculations of phonon-mediated superconducting properties are based on the Bardeen-Cooper-Schrieffer (BCS) theory. [58] There are three main approaches, namely i) semi-empirical methods based on the McMillan formula; [58] ii) first-principles Green's function methods based on the Migdal-Eliashberg (ME) theory [59,60] ; and iii) the density functional theory for superconductors (SCDFT). [61] Approaches (i) and (ii) are implemented in the EPW code. The electronphonon scattering process in the system is expressed by a discrete matrix elements: www.advancedsciencenews.com www.ann-phys.org where Ψ km+q and Ψ kn are the electronic eigenstates and q V stands for the linear change of the potential felt by the electrons due to the atomic displacement associated to a phonon with a branch index and wavevector q.
The phonon linewidth provides another way to gain experimental information about the coupling strength. The finite linewidth or inverse lifetime of a phonon mode is connected to the imaginary part of the phonon self-energy in the following equation: where q is the phonon energy, Ω BZ corresponds to the Brillouin zone, while kn and F set for electron energies in a single band n and Fermi level, respectively. This expression contains the temperature dependence via the Fermi distribution functions. Because phonon energies are typically small compared to electronic energies, it follows that measurements of the phonon linewidths, for example, by inelastic neutron or X-ray scattering experiments, provide information about the importance of phonon mode for the pairing.
The Eliashberg spectral function is given in terms of q as follows: [62] where N(E F ) presents electron density in the Fermi level, can be expressed as: with q sets for a local electron-phonon strength. The cumulative electron-phonon coupling strength is expressed by: where the dimensionless prefactor 2 F( ) can be interpreted as a measure of the coupling. Due to an individual phonon mode, the Eliashberg function is a sum over all phonon branches and averaged over phonon momentum. The logarithmic average frequency is given by: After obtaining the total and the ln , the superconducting transition temperature T c can be calculated by the pseudo empirical McMillan equation further modified by Allen-Dynes [63] and written in terms of the screened coulomb repulsion constant * as follows. According to ref. [64] , the superconducting properties are obtained from the self-consistent solution of the fully isotropic Migdal-Eliashberg equations on the imaginary axis at the fermion Matsubara frequencies j = (2j + 1)T (with j an integer) for each temperature T.
Using the standard approximations of Migdal-Eliashberg theory, [65] the two non-linear equations solved self-consistently are expressed as: and where the Z(i j ) is the mass renormalization function and Δ(i j ) is the superconducting gap defined as the energy necessary to break the bond of Cooper pairs.

Results and Discussion
In this section, we investigate the electronic and phononic properties of B 3 S and B 3 Se monolayers. Then, we deeply study how phonon-mediated attraction between electrons near the Fermi surface creates a Cooper pair and causes superconductivity in these two compounds. We start by reproducing the structural parameters of B 3 S displayed in the Figure 1. Next, we explore the compound B 3 Se for the first time. Optimization of the crystallographic structure reveals a perfectly planar honeycomb lattice without any buckling, however, the hexagonal rings are distorted. Due to the asymmetry of the lattice and the difference of atomic radius, the deformity of hexagonal shape is more pronounced in B 3 Se with respect to B 3 S. The optimized constants of the orthogonal primitive cells are a = 6.10 Å, b = 5.25 Å for B 3 S manolayer, in a good accordance with ref. [31] and a = 6.60 Å, b = 5.75 Å for germanium monosulfide sheet (B 3 Se). The unit cell consists of four atoms, namely, one X atom (with X=S, Se) and three boron atoms as shown in Figure 1a,c. It follows that in the sheet, each X atoms is surrounded by three boron atoms, while the first nearest neighbors of a B-atom are two B-atoms and one X-atom. In these anisotropic structures, the interatomic distances within the unit cell are about d BB ≃ 1.66 Å and d BS ≃ 1.84 Å for B 3 S, which are comparable to some graphene-based monolayers such SiC (d Si C = 1.78 Å) [14] and GeC (d GeC =1.965 Å), [15] while d BB ≃ 2.67 Å and   To examine the dynamical stability of these structures, we first examine the corresponding phonon dispersion plotted in Figure 2 for a 2× 2 supercell. The 24 bands detected for each system include three acoustical branches and 21 optical modes. No imaginary frequencies are observed along the high-symmetry direction of the Brillouin zone, confirming the perfect planar stability of both systems, which is an accurate and highly credible result, especially from the electron-phonon matrix elements. Notice also that the linear dependence of the acoustic branches near the Γ-point is a signature of hexagonal lattices. [66] The anisotropy of the lattice leads to a different speed of sound along the Γ -Z and Y -Γ directions. The highest recorded optical phonon frequency is 1200 cm −1 in B 3 S and 1037 cm −1 for B 3 Se.
Furthermore, energy stability analysis reveals a formation energy value of -3.86 eV per atom for B 3 S, and a value of -3.69 eV per atom for the B 3 Se sheet. Similar stable monolayer chalcogenides have been recently studied like VS 2 , which shows a higher formation energy (-4.94 eV per atom), [67] single-layer h-ZnX sheets (with X = S, Se, and Te) constructed from the cleavage of their bulk wurtzite phases, [68] as well as other hexagonal binary compounds such as planar osmium carbide monolayer. [20] The stability of these two materials is further evaluated from the perspective of molecular dynamics at finite temperatures 300 K (room temperature), 500 K, and 773 K, which corresponds to   The results are in agreement with some 2D planar honeycomb lattices, including h-BN, SiC, and GeC monolayers. [69] It follows that the high structural rigidity of our systems confirms their potential fabrication.
The band structures describing B 3 X with (X = S, Se), in Figure 4, are calculated using the local density approximation. Both systems exhibit metallic character where the conduction bands cross the Fermi level. A degenerate band intersects E F in the Z-T direction of the Brillouin zone. In addition, the B 3 S bands display four intersections with the Fermi level along the T-Γ direction with respect to only two intersections for B 3 Se. The result concordes well with the electronic band structure of hexagonal XB 6 (with X = Ga, In) monolayers [70] ; however, an www.advancedsciencenews.com www.ann-phys.org indirect gap semiconductor behavior is reported for P 3 Ga and P 3 Sn compounds. [71,72] In order to obtain accurate information about the energy and vibrational modes involved in the electron phonon coupling strength (EPC), Figure 5 shows the phonon linewidth derived from the phonon self energy mapped onto the phonon dispersion along the Brillouin Zone (BZ). This physical quantity is the main indicator of the creation of the EPC for each band in the phonon dispersion. Furthermore, the linewidth quantifies the scattering process in the EPC, which plays a key role for the following calculations, especially the Eliashberg spectral function.
A general analysis of the curves reveals that the phonon linewidth energy in B 3 S is higher than in B 3 Se. More precisely, it is greater than 6 meV for B 3 S versus 2.5 meV observed for B 3 Se. In addition, the longitudinal and transverse optical modes (LO and TO) stand behind the main part of the total phonon linewidth energy. Acoustic phonons exhibit a small phonon linewidth as expected and inferred from other systems like aluminene and stanene. [37,46] In compound B 3 S, the linewidth associated with the out-of-plane transverse and longitudinal optical modes (ZO, TO, LO) significantly exceed the maximum phonon linewidth in BN monolayer [73] and MoS 2 . [74] Also, B 3 Se compound exhibits much higher phonon linewidth compared to the tungsten diselenide WSe 2 . [75] The phonon density of states (PhDOS), plotted in Figure 6, peaks with fewer states for B 3 S compared to B 3 Se. The highest phonon density is 0.11 states detected at 750 meV for B 3 S and 0.16 located at 200 meV for B 3 Se. Moreover, the partial PhDOS shows a dominant contribution of boron atoms in both systems and a presence of sulfur and selenium atoms at lower frequencies. Our result is in good agreement with ref. [76]. It should be noted that this key parameter (PhDOS) for the electron and phonon eigenenergies and indirectly for the relaxation time interferes significantly with carrier transport for electrical conduction and also heat transfer. The presence of phonons of more than 1000 meV in both B 3 S and B 3 Se compounds gives them an enormous ability to be considered as promised heat conductors like the h-BN monolayer [77] or as excellent thermoelectric materials such as SnP 3 . [78] On the other hand, two pockets are observed in the T-Z axis of the Fermi surface of B 3 S, with a slight pocket around the Ypoint. Figure 7 also shows that the two circular pockets are more centered in the T-point, for the B 3 Se compound, with a slight presence on the Z-point. The absence of the electronic density around the Γ-point is related to the intersection of the bands with the Fermi level as previously reported in Figure 4. This result evaluates the electronic density in the Fermi surface which is higher in B 3 S with respect to the B 3 Se monolayer. This finding is very important due to the involvement of the Fermi level as part of the electronic transition given in Equation (4) that proves its role in superconductivity, in accordance with ref. [79] relating how the existence of electronic density in the Fermi level of the metallic SnH 4 material provides an environment for enhanced electron-phonon coupling. Furthermore, the band intersections with the Fermi level interferes highly with the density of states leading to the ability of existence of electrons in this part. This should contribute significantly to the electronic [80] and superconducting aspects [81] of our materials as previously mentioned, and it should also affect the electric properties such as the resistivity given by the Ziman formula based on the Migdal-Eliashberg transport spectral function. [82] To explore the contribution of each single frequency in creating the electron-phonon coupling strength, we illustrate the Eliashberg spectral function 2 F( ) and the cumulative frequencydependent EPC function ( ) in Figure 8. The highest peaks of 2 F( ) are observed in the lowest frequencies which are basically at the same frequency corresponding to the total PhDOS presented in Figure 8. Furthermore, the cumulative electronphonon strength ( ) is related to the spread of the peaks due to the integral calculation in Equation (5), and it is an indicator of the superconductivity of B 3 S and B 3 Se. The result is in accordance with the previous PhDOS values of 1.2 for B 3 S and 0.82 for B 3 Se, which are strong and can quantify the decay capacity in these materials. More particularly, the electron-phonon coupling in B 3 S can be qualified as high as for the case of the triangular borophene B ▵ having 1.1 [83] and the monolayer W 2 N 3 exhibiting a high EPC of 1.75. [84] However, the EPC is more medium for B 3 Se and similar to that of CaB 6 , which is equal to 0.87, [85] but it is higher than 0.5 detected for Li-decorated graphene. [86] The integrated electron-phonon coupling strength is presented along the high symmetry path of Brillouin zone in Figure 9. For B 3 S, the e-p strength is concentrated around the Γ − point, which presents the largest EPC as already reported earlier. In contrast, the EPC distribution is quite scattered for compound B 3 Se, with its largest amount detected around the Y-point. One can deduce that the zone exhibiting higher electron-phonon www.advancedsciencenews.com www.ann-phys.org  coupling corresponds to the one with less electronic density in the Fermi surface as shown in Figure 7. Specifically, the electrons in the Fermi level in Figure 7 are in their rest position before the interaction with phonon energy as explained for the monolayer biphenylene [87] and the Kagome metal CsV 3 Sb 5 . [88] Therefore, the zone of electron decay of the Fermi surface is the most vacuumed in terms of density as shown in Figure 7. One can also observe that the maximum EPC reaches 0.75 in B 3 S, while it touches 0.42 in B 3 Se, and there also exists a slight presence of EPC toward the four directions close to Z point. It follows that the B 3 S monolayer can be a better superconductor with higher critical temperature compared to its counterpart B 3 Se.
The main indicator on superconductivity is the critical temperature, which separates the behavior of the material between superconductivity and the normal state by using the Allen-Dynes formula under the Coulomb screened constant * = 0.10 used mainly in the literature. [46,89] The critical temperature for B 3 S is 20 K while it is 10.5 K for B 3 Se. Therefore, the T C of B 3 S is twice bigger than that of B 3 Se and is higher than the 13.2 K reported for the calcium-decorated hexagonal boron nitride monolayer, [90] Li-decorated graphene showing a T C = 5.9 K, [91] Ca-intercalated bilayer graphene exhibiting phonon-mediated superconductivity with a T C in the range [6.8 − 8.1] K, [92] and GaB 6 presenting 14.5 K. [93] To examine the effect of the Coulomb pseudopotential on the variation of the critical temperature, Table 1 lists T C at different values of * . It can be seen that T C reduces slightly as * increases. It is worth noting that the value of the effective screened Coulomb repulsion constant is generally chosen between 0.1 and   0.15 for 2D materials; consequently, our superconducting gap calculations are set to * = 0.10 to simplify the comparison between different materials. [94,95] Figure 10 displays the superconducting gap (Δ) which corresponds to the binding energy of the Cooper electron pair. In contrast to multigap superconductors, both B 3 S and B 3 Se exhibit a Figure 11. Distribution of single superconducting gap on k-space at absolute temperature 0.1 K.
single anisotropic superconducting gap since the metallic property of these 2D materials arises only from the delocalized p z orbitals of the B-atoms perpendicular to the molecular plane. Near the Fermi level, these p z -orbitals create conjugate bonds for the conduction electrons in good agreement with what is reported in ref. [31] . Furthermore, Figure 10 reveals that the anisotropic single superconducting gap is smaller against increasing temperature which concordes with previous works, especially those dealing with the alkaline earth hydride CaH 6 [96] and the noncentrosymmetric superconductor PbTaSe 2 showing a value around 0.3 meV. [97] A zoom on the gap energy distribution near the high symmetry points is plotted in Figure 11 at a fixed temperature of 0.1 K. In order to evaluate the ideal ratio given by the BCS theory, the center of the peaks, shown in Figures 10 and 11, is taken as the point for each calculated gap energy. Consequently, the single superconducting gap values Δ decrease from 3.5 meV for B 3 S and 1.8 meV for B 3 Se estimated at 0 K until reaching the critical phase which splits the superconducting phase of the system from its normal phase by breaking the Copper superfluid couple; the electrons then move separately in the structure. The ideal ratio, given by the relation (Δ∕k B T c = 1.76 ), results in the ratio of 1.85 for B 3 S and 1.77 for B 3 Se. These values are in agreement with the FeSe monolayer. [98]

Conclusion
Using density functional perturbation theory, two new superconducting chalcogenides are predicted, namely discovered B 3 S and B 3 Se. The examination of the structural parameters as well as phonon dispersion provides the dynamical stability of both systems which is required for their possible fabrication. The band structure reveals metallic character for the two boron-based chalcogenide monolayers. The electron-phonon coupling is evaluated to be 1.2 for B 3 S and 0.82 for B 3 Se. Furthermore, the decay process of the electron-phonon pairing is quantified by investigating the phonon and electron densities of states and phonon linewidths. The critical temperatures for B 3 S and B 3 Se are found to be 20 and 10.5 K, respectively, which promote the integration of these new 2D nanomaterials in future potential superconducting applications