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Electron-phonon coupling superconductivity in 2D orthorhombic MB6 (M=Mg, Ca) and hexagonal MB6 (M=Mg, Ca, Sc, Ti, Sr, Y)

Combining crystal structure search and first-principles calculations, we report a series of two-dimensional (2D) metal borides including orthorhombic (ort-) MB6 (M=Mg, Ca) and hexagonal (hex-) MB6 (M=Mg, Ca, Sc, Ti, Sr, Y). Then, we investigate their geometrical structures, bonding properties, electronic structures, mechanical properties, phonon dispersions, thermal stability, dynamic stability, electron-phonon coupling (EPC), superconducting properties and so on. Our ab initio molecular dynamics simulation results show that these MB6 can maintain their original configurations up to 700/1000 K, indicating their excellent thermal stability. All their elastic constants satisfy the Born mechanically stable criteria and no visible imaginary frequencies are observed in their phonon dispersions. The EPC results show that these 2D MB6 are all intrinsic phonon-mediated superconductors with the superconducting transition temperature (Tc??) in the range of 2.2-21.3 K. Among them, the highest Tc (21.3 K) appears in hex-CaB6, whose EPC constant () is 0.94. By applying tensile/compressive strains on ort-/hex-CaB6, we find that the compressive strain can obviously soften the acoustic phonon branch and enhance the EPC as well as Tc. The Tc of the hex-CaB6 can be increased from 21.3 K to 28 K under compressive strain of 3%. These findings enrich the database of 2D superconductors and should stimulate experimental synthesizing and characterizing of 2D superconducting metal borides.

and should stimulate experimental synthesizing and characterizing of 2D superconducting metal borides.

I. INTRODUCTION
Superconducting materials have excellent physical and chemical properties, and may be widely used in electric power, electronics, medical, transportation, high-energy physics and other fields. In addition to the hot topics of copper oxide [1] and iron-based superconductors [2], conventional electron-phonon superconductors are still attractive [3]. Among reported conventional phonon-mediated superconductors, the anisotropic two-gap superconductor MgB2 ever holds the record of the highest superconducting transition temperature (Tc) of 39 K [4]. This discovery has stimulated many efforts in exploring the microscopic mechanisms [5][6][7] and also in searching new conventional superconductors [8,9]. The predicted superconductors of hydrogen sulfide [10] and LaH10 [11], based upon the Bardeen-Cooper-Schrieffer (BCS) theory [12], have been experimentally verified [13][14][15]. Their Tc of 203 and 250 K have greatly overcome the Tc of MgB2.
The superconductivity with high Tc may exist in materials composed of light atoms [43]. Boron has a very small atomic mass, which may enable superconducting behaviors in its elemental or alloying forms. Experimentally, a high Tc of 36 K was observed in boron-doped Q-carbon [8]. In bulk YB6, a Tc of~7 K was reported through measuring its specific heat, resistivity, and magnetic susceptibility [44]. Till now, superconducting properties have never been experimentally observed in 2D boron and 2D borides. The difficulty may in the synthesizing and also in measuring.
Recently, we also predicted some novel 2D superconductors of Mo2B2 [33], W2B2 [34], and XB6 (X=Ga, In) [45] based on the crystal structure prediction method and first-principles calculations. A certain tensile/compressive strain can regulate the EPC as well as Tc of these materials. In the present work, we first report a series of 2D MB6 structures including orthogonal (ort-) MgB6, CaB6, and hexagonal (hex-) MgB6, CaB6, ScB6, TiB6, SrB6, and YB6 through structural searching by using the CALYPSO code.
Then, we investigate their crystal structures, electronic structures, phonon dispersions, mechanical properties, EPC, and superconducting properties by performing first-principles calculations. We find that these 2D metal hexaborides are all phonon-mediated superconductors with Tc in range of 2.2-21.3 K. The underlying mechanism of their superconductivity is carefully analyzed. In addition, the effect of tensile/compressive strain on the EPC constant and superconducting properties is also studied.

II. COMPUTATIONAL DETAILS
We search for 2D MB6 (M= Mg, Ca, Sc, Ti, Sr, Y) using the particle swarm optimization (PSO) scheme as implemented in the CALYPSO code [46][47][48]. In searching, both planar and buckled structures including one, two, and three layers are considered. For each MB6 system, one, two, and four formula units per simulation cell are calculated. The number of generations are set to be 30 and each generation contains 30 structures. After structural searching, thousands of 2D MB6 structures are generated. The subsequent structural optimization, energy calculations, and ab initio molecular dynamics (AIMD) simulations are carried out by utilizing the density functional theory (DFT) method as implemented in the Vienna Ab initio Simulation Package (VASP) [49]. The Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) [50] is employed and the electron-ion interaction is described by using the projector augmented wave (PAW) method [51]. The plane-wave cutoff energy is set as 600 eV and the Brillouin zone (BZ) integration is performed using a 11  11  1 Monkhorst-Pack (MP) mesh. All the geometries are relaxed until all forces on the atoms are smaller than 0.005 eVÅ -1 . A vacuum separation is set to more than 20 Å to prevent any interactions between two neighboring monolayers. The van der Waals (vdW) correction is included by using the optB86b-vdW exchange functional [52] since it can properly treat the long-range dispersive interactions. The AIMD simulations in the NVT ensemble are employed at 300, 700, and 1000 K to investigate the thermal stability of these 2D MB6 monolayers.
They are performed by using 3  3 supercells with a time step of 1 fs and total simulation of 5 ps.
The calculations of band structure, phonon dispersion, EPC, and superconducting property are performed at the DFT level, employing the norm-conserving pseudopotentials [53] as implemented in the QUANTUM-ESPRESSO (QE) package [54]. The VASP-optimized structures are re-optimized within QE. The plane-waves kinetic-energy cutoff and the energy cutoff for charge-density are set as 80 and 320 Ry, respectively. The Methfessel-Paxton smearing width of 0.02 Ry is used. The BZ k-point grid of 32321 is adopted for the self-consistent electron density calculations.
The dynamic matrix and EPC matrix elements are calculated on a 8  8  1 q-point mesh for both ort-and hex-MB6 monolayers. The phonon properties and EPC are calculated within the density-functional perturbation theory [55] and Eliashberg theory [56].
The EPC qv  is calculated according to the Migdal-Eliashberg theory [57] by where BZ  is the volume of BZ, ϵkn and ϵk+qm indicate the Kohn-Sham energy, and , v kn k qm g  represents the EPC matrix element. The Eliashberg spectral function α 2 F(ω) can be estimated by Then, the EPC constant  (ω) and the logarithmic average frequency ωlog(ω) can be determined by After obtaining the total  and ωlog, the superconducting transition temperature Tc can be calculated by In our following calculations, a typical value of the effective screened Coulomb repulsion constant µ* = 0.1 is used [24,58]. corresponding hexagonal phase, respectively. In one recent theoretical work [32], the energy of the ort-MgB6 was also reported lower by 1.02 eV per unit cell than that of the hex-MgB6. The global minimum structure of the 2D AlB6 [35] is also the orthogonal one. However, the phonon spectra results calculated by VASP as well as QE show that the orthogonal structure of ScB6, TiB6, SrB6, and YB6 is dynamically unstable. As known, the global minimum energy is not the exclusive condition for experimental synthesis of materials. Under suitable conditions, it is possible to synthesize some metastable crystals with higher energy [59]. On the basis of aforementioned discussions, in the following, we will focus on the (meta)stable  Table I To get more information about the bonding, we calculate the line charge density distribution along the nearest B-B and M-B bonds and perform the electron localization function (ELF) [62] and Bader charge analysis [63]. The ELF (Fig. S1) pictures as well as the values of Bader charge (QB) and charge density at the corresponding bond point (CDb) (Table S1) are presented in the supporting information (SI). For the ort-/hex-MgB6, hex-ScB6, and hex-TiB6, there are mainly covalent bonds between B atoms and ionic bonds between metal atoms and the neighboring B atoms. However, for the ort-/hex-CaB6, hex-SrB6, and hex-YB6, in addition to the above chemical bonds, there are also metallic bonds between M atoms.

III. RESULTS AND DISCUSSION
One can see the detailed discussions in the SI.

B. Stability and mechanical properties of the ort-and hex-MB6
It is important to verify the stability of these MB6 monolayers, which is helpful for judging whether they can be synthesized experimentally. Cohesive energy is a widely accepted parameter used to evaluate the stability of materials. We calculate the cohesive energy Ecoh using the following formulas: where EM and EB are the total energies of the isolated metal and B atoms, respectively, E 6 MB is the total energy of the 2D MB6. As shown in Table I, the calculated cohesive energies are 5.42, 5.55, 5.27, 5.41, 6.00, 6.32, 5.28, and 5.93 eV/atom for the ort-MgB6, ort-CaB6, hex-MgB6, hex-CaB6, hex-ScB6, hex-TiB6, hex-SrB6, and hex-YB6, respectively. These values are larger than those of the previously computationally predicted and later experimentally fabricated 2D materials, such as Cu2Si (3.46 eV/atom) [64,65], Ni2Si (4.80 eV/atom) [66], and Cu2Ge (3.17 eV/atom) [67]. The relatively large cohesive energies indicate that these MB6 monolayers can be synthesized under appropriate experimental conditions. We also calculate the phonon dispersions along the high-symmetry lines as well as the phonon density of states (PhDOS) to evaluate the dynamic stability of these MB6 monolayers. For the ort-and hex-MgB6, ort-and hex-CaB6, hex-ScB6, and hex-TiB6, the fact that no imaginary modes in the BZ confirms the lattice dynamic stability of these six MB6 monolayers. A detailed discussion of the phonon dispersions for all the eight MB6 monolayers will be supplied later. In addition, thermal stability is very important for the practical application of these 2D MB6 in nano-electronic devices or electrode films. In order to investigate the thermal stability of these 2D MB6, AIMD simulations are performed with total simulation time of 5 ps at 300, 700, and 1000 K. We present in Fig. S2, S3, and S4 the variation of total energies with time for the MB6 monolayers. The snapshots after 5 ps for these simulated MB6 are shown in Fig. S5 and S6. We find that the total energy of each MB6 monolayer fluctuates around a constant value at 300 and 700/1000 K. After simulating for 5 ps, the framework of each MB6 monolayer well maintains its original configuration at both 300 and 700/1000 K. The above results indicate the excellent thermal stability of these MB6 monolayers.
Mechanical properties are also very important for the practical application of 2D materials. In order to accurately understand the mechanical properties of these MB6 monolayers, we calculate their elastic constants [68]. Generally speaking, a mechanically stable 2D material needs to satisfy the Born criteria [69], C11C22 -C12 2 > 0 and C66 > 0. For hexagonal symmetry, C11 = C22. Thus, the stability criteria is |C11| > |C12| and C66 > 0. The calculated elastic constants of these eight MB6 monolayers are listed in Table II. We find that the elastic constants of these MB6 monolayers all meet the Born criteria, indicating that they are mechanically stable. Based on the elastic constants, the Young's moduli along the x and y directions of these materials are calculated according to 11 22 12 21 22 ( )   there are several Dirac cones above the Fermi level, close to the high-symmetry point S in the first BZ, which is similar to that of the ort-AlB6 [35]. There are many bands crossing the Fermi level, which forms five pockets at the FS. As shown in Fig. 3(a), there are two pockets centered at X and Y points, respectively, one pocket along -X line, and two pockets along X-S line. For ort-CaB6, there are one, three, one, and one band(s) crossing the Fermi level along -X, X-S, S-Y, and Y-, respectively. As shown in Fig. 3(b), these bands form one dumbbell-shaped pocket centered on X, one elliptical pocket centered on Y, and one pocket along X-S. For hex-MgB6, at the K point, there is one Dirac cone across the Fermi level whose states are mainly from the 2p orbitals of the B kagome lattice. This phenomenon has also been observed in one recent theoretical work [61]. Actually, kagome 2D lattice has been extensively studied owing to its unique physics related to Dirac bands [72]. In order to investigate the EPC and superconducting transition temperature Tc of these 2D MB6, we calculate their phonon dispersions, PhDOS, Eliashberg spectral function α 2 F(ω), and λ(ω) and present them in Figs. 4, 5, and 7. As shown in Fig. 4 and Fig. 5, there are no imaginary phonon modes in the phonon dispersions for ort-MgB6, ort-CaB6, hex-MgB6, hex-CaB6, hex-ScB6, and hex-TiB6. This clearly indicates that these materials are dynamically stable. The EPC integrated over all phonon branches and distributed in the BZ are shown in Fig. 6 for these six materials.
Our calculated phonon spectra for the ort-MgB6 are consistent with those presented in Liao's work [32]. From our decomposed phonon dispersions, we find that the Mg vibrations and the out-of-plane Bz vibrations dominate the low frequencies below 500 cm -1 , while the in-plane Bxy vibrations dominate the high frequencies above 500 cm -1 .
The low-frequency phonons (150-350 cm -1 ) around X point, mainly associated with the Bz vibrations, contribute 68% of the total EPC (λ=0.56). As shown in Fig. 6(a), the region around the X point possesses the largest EPC, which is consistent with the EPC λqv in Fig. 4 For hex-SrB6 and hex-YB6, as shown in Fig. 7, there exist imaginary frequencies in their BZ. However, such small negative vibrations can be understood by the Kohn anomaly [73]. The Kohn anomaly of the phonon mode always yields significant coupling between electrons and acoustic phonons and may induce large EPC [24,31]. To have a more comprehensive view of our predicted 2D MB6, we compare their EPC constants λ and Tc with the reported borophenes [24,26] and 2D metal borides [32][33][34][35]45]. Among all the materials listed in Table III, our studied hex-CaB6 possesses the largest Tc (21.3 K). In addition to hex-CaB6, our obtained hex-SrB6 also have large Tc of 15.9 K, which is larger than that of B ◇ [24], β12 [26], ort-AlB6 [35], ort-/hex-GaB6 [45], ort-/hex-InB6 [45], Mo2B2 [33], and W2B2 [34], while smaller than that of B △ /B □ [24]. The Tc of our studied materials, except for hex-TiB6, is greater than that of ort-AlB6 [35]. Compared with bulk YB6 [44,74], 2D hex-YB6 has larger N(EF) and EPC constant, which leads to a larger Tc. The different arrangement of atoms in the crystal cell results in the difference of superconductivity between 2D and bulk YB6.
The details and colors are the same as in Fig. 4. a This is experimental data. When these 2D MB6 are grown on substrates, strains will be introduced, therefore, the superconducting properties should be affected. Therefore, we apply strain by changing the lattice parameters to investigate the effect of strains on the EPC and superconducting properties of these 2D materials. Here, we choose the ort-CaB6 and the hex-CaB6 as the representatives to study. First, we discuss the dynamical stability of the ort-CaB6 and the hex-CaB6 under different strains. The equi-biaxial tensile and compressive strains have been applied on these two configurations. The positive value of ε represents tensile strain while the negative value represents compressive strain.

E. Strain engineering
For ort-CaB6, we find that it is dynamically stable under the strain region of -1% ≤ ε ≤ 3%, indicating that it can withstand up to 1% compressive strain and 3% tensile strain.
The phonon dispersions under ε = -1%, 0, and 3% are displayed in Fig. 8(a). We find that the compression strain makes the phonon frequency at low frequency (< 500 cm -1 ) lower down, but those at high frequency (> 500 cm -1 ) higher up. When tensile strain is applied, the trends of the phonon frequency change are opposite. As shown in Fig. 8(b) and (c), when the strain ε increases from -1% to 3%, the logarithmically averaged phonon frequency ωlog increases linearly, whereas the N(EF), the EPC constant , and the Tc decreases linearly. Under compressive strain of -1%, the largest Tc of 12 K is obtained.
For hex-CaB6, it is dynamically stable under the strain region of -3% ≤ ε ≤ 2%, indicating that it can bear more compression strain than ort-CaB6. The phonon dispersions under ε = -3%, -2%, 0, and 2% are displayed in Fig. 9(a). Like ort-CaB6, the compression strain makes the phonon frequency at low frequency (< 500 cm -1 ) lower down, but those at high frequency (> 500 cm -1 ) higher up. As shown in Fig. 9(b) and (c), when the strain ε increases from -3% to 2%, the logarithmically averaged phonon frequency ωlog increases linearly, whereas the N(EF), the EPC constant , and the Tc are almost linearly reduced. Under the maximum compressive strain of ε = -3%, the Tc can be modulated up to 28 K. Therefore, the compressive strain can strengthen the EPC and increase the Tc of both ort-CaB6 and hex-CaB6.

IV. CONCLUSION
In our present work, by combining the crystal structure search as well as systematic first-principles calculations, we predict a series of 2D metal borides, Our findings in the present work enrich the database of 2D metal borides and may stimulate experimental synthesis of these materials and also 2D metal carbides and nitrides. The superconducting properties of these 2D MB6 supply experiments another choice other than the 2D boron to realize superconducting monolayers in boron based 2D forms. After all, the thickness of these 2D MB6 is prior to borophene and their Tc are relatively high.

NOTES
The authors declare no competing financial interest.