Superconductivity in two-dimensional MB4 (M = V, Nb, and Ta)

The study of two-dimensional (2D) superconductors is one of the most prominent areas in recent years and has remained a long-standing scientific challenge. Since the introduction of the new member of 2D material family consisting of transition-metal elements with Borides (MBenes), the study of various properties of 2D metal borides has attracted widespread interest. In this work, we systematically investigate the phonon-mediated superconductivity in 2D MB4 (M = V, Nb, and Ta), by using the first-principles calculations. Starting from the dynamical stabilities of these structures, we perform a detailed analysis of the electron–phonon coupling and superconductivity of AB-stacked MBenes by solving the anisotropic Eliashberg equation. NbB4 has the largest electron–phonon coupling and the highest superconducting transition temperature T c of 35.4 K. Our study broadens the 2D superconducting boride family, which is of great significance for the study of 2D superconductivity.


Introduction
Two-dimensional (2D) materials, one of the most widely studied materials, have a broad application prospect in future electronic devices, because the quantum confinement effect leads to unique properties in the monolayer limit [1].The research of 2D materials is of great significance to innovating fundamental physics and material science.Since the successful exfoliation of graphene in 2004 [2], a large number of interesting 2D materials have been predicted theoretically, such as MoS 2 [3,4], silicene [5,6], germanene [7], phosphatene [8], transition-metal dichalcogenides [9,10], etc. 2D materials consist of metals, semi-metals, insulators, and semiconductors with direct or indirect bandgap [11][12][13][14].With the rapid development of nanotechnology, many 2D materials are also synthesized in experiments [15][16][17].And 2D boron sheet, namely borophene [18], has various structures because of the outer-shell electron configuration of boron elements.They have been prepared in different experimental research groups [19,20].In contrast to other monolayer materials, borophene is not stripped from the bulk layered counterparts, but is directly grown on the metal substrates [19] and the free-standing form is yet to be obtained.The borophene possesses abundant carrier density and higher Young's modulus than graphene [21].And the delocalized multicenter bonding results in the metallic conductivity of borophene, different from the bulk boron [22,23].The excellent mechanical properties and controllable anisotropic electronic properties make borophene a good candidate for future development prospect [24,25].
Superconducting materials have long been an important research field of fundamental physics and materials science.They have wide application prospects in many fields, due to their zero resistance and complete diamagnetism below the superconducting transition temperature [26,27].Recently, the combination of superconductivity and 2D materials not only offers an experimental platform to discover novel physical properties in 2D limit, but also provides further theoretical insights to the origin of high temperature superconductivity.Unfortunately, graphene is difficult to show good intrinsic superconductivity because the density of electronic states near the Fermi level is zero.However, it has been reported that inserting metallic atoms into graphene can lead to superconductivity [28][29][30].For example, calcium inserted in bilayer graphene (C 6 CaC 6 ) is proved to show superconductivity below 4.0 K [31].The small atomic mass of boron also suggests high Debye frequency, which can effectively promote superconductivity [32].Thus, the phonon-mediated superconductivity in borophene has attracted great interest [30,[33][34][35].Penev et al found that the superconducting critical temperature (T c ) of borophene (10.0-20.0K) increases with the decrease of vacancy number [32].Zhao Y et al also reported that several structures of borophene show superconductivity above the liquid hydrogen temperature [34].These results suggest that borophene may show the highest T c among all pure single-element materials in the absence of external high pressure and strain [32,36].
Inspired by the graphene adsorbed by metal [30], the addition of a metal layer to the boron atomic layer proposed in recent years is a new member of the 2D material family known as MBenes [37].MBenes are previous transition metal borides and the latest derivatives are the ternary or quaternary transition metal boride phase.Excellent mechanical, metallic/semiconducting, and magnetic properties motivate MBenes to be used for designing promising next-generation electronic devices, energy storage, and so on.In addition, a few previous reports hinted that MBenes might be superconductors [35,[38][39][40][41]. Thus, the search for high T c superconductors in 2D boride has aroused our great interest.In this work, we study the electron-phonon coupling (EPC) and superconducting properties of MB 4 , AB-stacked bilayer δ 3 borophene with intercalated metal atom (M = V, Nb, and Ta).The largest EPC exists in NbB 4 with the highest T c of 35.4 K.We also find that biaxial strain increases T c for VB 4 and TaB 4 while it suppresses the superconductivity in NbB 4 .

Computational methods and details
The first-principles calculations were implemented with Optimized Norm-Conserving Vanderbilt pseudopotentials in QUANTUM ESPRESSO package [42,43].The exchange related functions were described by local density approximation.MB 4 was simulated with a vacuum thickness of 20 Å, to eliminate the influence of the periodicity of the crystal structure.To guarantee that the structure is fully optimized, the planewave kinetic-energy cutoff and the energy cutoff for charge density were set to 50 Ry and 500 Ry, respectively.The k-point grid of the wave vector space integration was set to 24 × 24 × 1 and electronically smeared in the methyl container-Paxton scheme.Based on these parameter settings, the force on each atom was kept below 10 −5 Ry/bohr and the total energy varied by less than 10 −5 Ry/bohr during structure optimization.The electron charge density was calculated at the Brillouin-zone (BZ) k point mesh of 24 × 24 × 1.The phonon modes were computed in the density functional perturbation theory [44] at q point mesh of 12 × 12 × 1.The maximally localized Wannier functions required in the EPW calculation were interpolated by using the BZ k-mesh of 24 × 24 × 1 [45,46] and the s and p orbitals of B and the s orbitals of the metal atoms were chosen as the projectors.Subsequently, for the dynamically stable crystal structures, the solutions of the fully anisotropic Migdal-Eliashberg equation were calculated with interpolated k-point grids and q-point grids of 120 × 120 × 1 and 120 × 120 × 1, using EPW codes [47][48][49].The degaussw and degaussq were set to 0.015 eV and 0.2 meV, respectively.Thus, the corresponding electron-phonon coupling and superconducting properties of each structure were obtained.Finally, we tried biaxial tensile or compressive strain to increase the superconducting transition temperature.

Results
According to the definition of coordination number (CN) to classify boron monolayer sheets into various types, a single CN is named δ and the hexagonal honeycomb crystal structure has CN = 3, so the graphene-like borophene is marked as δ 3 .Our calculation shows the instability of δ 3 borophene (figure S1).Therefore, the three transition metals, V, Nb, and Ta in group V, are considered as intercalation of bilayer δ 3 borophene in order to have the structural stability.Figure 1 plots the top and side views of MB 4 (space group: P3m1), wherein the metallic atom is located in the hexagonal center of the lower layer, aligned with the upper B atom.The interaction between M and B atoms in the upper layer induces the wrinkled atomic layer in the structural optimization, as shown in figure 1(b).For the lower layer of AB-stacked MB 4 , the metal atoms are located in the center of the six-membered ring, and the metal atoms are evenly spaced with the three nearest neighbor B atoms, which does not destroy the flat structure of B atoms.While, for the upper layer, the metal atom is aligned with the one nearest neighbor B atom, it is easy to have a strong interaction, which leads to the wrinkles.Table S1 lists lattice parameters, the interlayer distances (d 1 and d 2 ), and the atomic fluctuation height (∆d).∆d of VB 4 , NbB 4 , and TaB 4 are 0.396 Å, 0.374 Å, and 0.388 Å, respectively.Although it is found that the AA-stacked MB 4 has a lower total energy than the AB-stacked structure, the previous work has found that the EPC constant (λ) of AA-stacked MB4 is very small [41], which is unfavorable to the superconductivity.Then, This work focuses on the AB-stacked MB 4 .After a molecular dynamic simulation at room temperature, the total energies always oscillate within a reasonable range without significant changes (figure S2).And the absence of imaginary frequency in the phonon spectra ensures that they are all  dynamically stable (figure 3).The calculations of the electronic structure show that the three structures all have electronic states at the Fermi level, showing metallic properties.Figure 2 shows the projected band structures and density of states (DOS) of MB 4 by the M-d orbital and B-p orbitals, respectively.NbB 4 has more electronic bands that cross the Fermi level than the other two cases, so it has the highest DOS at the Fermi level (3.25 states/eV/f.u.).The electronic states around the Fermi level are mainly contributed from the B-p orbital.To illustrate this point in more detail, we classify both p and d electron orbitals into in-plane and out-of-plane contributions.By comparing the relative weights of DOS at the Fermi level, we deduce that, for M = V or Ta, the in-plane and out-of-plane modes contribute equally.However, the out-of-plane states (p z ) of B atoms in NbB 4 have more significant contributions than the in-plane states.And figure S3 shows the projected band structures and DOS of MB 4 by two layers of boron atoms, respectively, it is clear that boron atoms in the flat layer contribute more to the Fermi level than boron atoms in the fold layer.
Next, we focus on two important aspects of the superconducting properties: the EPC constant (λ) and superconducting critical temperature (T c ). λ was obtained by solving the Eliashberg function [50], and T c was calculated by McMillian-Allen-Dynes formula [51,52]: where µ * is the retarded Coulomb pseudopotential which measures the strength of electron-electron interaction [53].And ω log is the logarithm average of the phonon frequencies: Figure 3 shows the calculated phonon dispersion, phonon DOS and Eliashberg spectral function α 2 F(ω) of MB 4 .The phonons of the three materials can be roughly divided into low-frequency and high-frequency regions.The former is mainly dominated by the vibration of metal atoms.With the atomic mass increases, the distinction between the vibration frequencies of metal atoms and B atoms becomes more obvious.More importantly, the phonons from B atomic vibrations contribute significantly to EPC and Eliashberg spectral function α 2 F(ω).By calculating the cumulative frequency-dependent λ(ω) of VB 4 (figure 3(a)), it found that phonons from 0 cm −1 to 200 cm −1 contribute only 18% to the total EPC strength, while the phonons in the range between 200 cm −1 and 600 cm −1 contribute about 67% of EPC.We also illustrate the atomic vibrational patterns for modes with strong coupling, as shown in figure 3(j).Two modes at 202 cm −1 and 406 cm −1 are the in-plane vibration of the B atoms in the upper layer and the out-of-plane relative vibration of the double layer, respectively.The mode at 466 cm −1 is out-of-plane relative vibration between B atoms and the V atoms.Similar results are found for the other two materials.For NbB 4 , the phonon with strong coupling is at around 121 cm −1 corresponding to the in-plane vibration of the B atoms in the upper layer (figure 3(j)).There are also several peaks in the range of 310-650 cm −1 in α 2 F(ω).The strong EPC of phonons with the frequency of 400 cm −1 at this point involves opposite vibrations of B of two layers the z direction.And the phonon at 564 cm −1 is the out-of-plane optical mode of two B atoms in the lower layer.Similarly, the phonons of B atoms vibration in TaB 4 also play a major role in EPC.For example, the mode at 143 cm −1 is the in-plane vibration of the upper B atoms, and the mode at 421 cm −1 corresponds to the in-plane vibration of the lower B atoms.Additionally, for all three structures, the optical phonon with the highest frequency (in-plane opposite vibration of B atoms in the lower layer) has fairly flat dispersion, which results in the peaks in the phonon DOS as well as α 2 F(ω).However, the high frequency makes a small contribution to EPC.Considering the isotropical approximation in the Allen-Dynes formula which may not be able to precisely describe the EPC and superconducting gaps, we used the EPW code [45,46] to resolve the fully anisotropic Migdal Eliashberg equations [46][47][48][49] in a self-consistent way to test the anisotropic effect.The band structure interpolated by the maximum localized Wannier functions (MLWFs) is consistent with the DFT calculation results (figure S4), which lays a substantial foundation for the next calculation.Firstly, anisotropic Eliashberg calculation shows that VB 4 is a two-gap superconductor, with the large superconducting gap stemming from the Fermi surface around Γ point and the small superconducting gap stemming from the Fermi surface around K point (figure 3(d)).As the temperature increases, the superconducting gap becomes smaller and vanishes at 13.6 K (figure 3 Then we consider the strain effect on the EPC and superconductivity, because strain engineering has been widely used in 2D materials to control the physical properties [54,55].The strains are introduced by the change of the lattice parameter ε = 100% × (a − a 0 )/a 0 , which does not destroy the dynamic stability of the structure (figures 4 and S5-S7).As shown in figure 4(a), the increase of the lattice constant reduces the frequency of the entire phonon spectrum, which is beneficial to increase the EPC constant.And the higher DOS at the Fermi level (table S2) leads to the higher peaks of α 2 F(ω), especially around 182, 532, and 822 cm −1 .Thus, the biaxial strain of ε = 5% can enhance T c of VB 4 from 13.6 K to 22.1 K (figure 4(b)).A similar effect can also be found when TaB 4 is applied with a tensile strain of ε = 5% (figure S6 and table S2).However, a slight decrease of DOS makes the enhancement less noticeable.For NbB 4 , the negative effect brought by the rapidly decreasing DOS (figure S5 and table S2) is obviously greater than the positive effect of the phonon frequency change under tensile strain, which results in a reduction of the T c value from 35.4 K to 28.2 K (figure S5).

Conclusion
In summary, we have studied the superconducting properties of AB-stacked MB 4 (M = V, Nb, and Ta) by using the first-principles calculations in combination with the Eliashberg formalism.We have proved that they all have superconducting properties and that NbB 4 has the best superconductivity with a T c of up to

Figure 1 .
Figure 1.(a) Top and (b) side views of AB-stacked MB4.The black lines indicate a primitive unit cell.The green and blue spheres denote boron and metal atoms, respectively.The perpendicular distance between the upper and lower B atoms and the metal atoms, and the atomic fluctuation height, denoted as d1, d2, and ∆d respectively.

Figure 2 .
Figure 2. The projected electronic band structures (a) VB4, (b) NbB4, (c) TaB4 and the corresponding DOS for M-d and B-p orbitals.

Figure 3 .
Figure 3. Phonon dispersion, phonon density of states (PHDOS), isotropic Eliashberg spectral functions α 2 F(ω), and cumulative frequency-dependent coupling λ(ω) are shown for (a) VB4, (b) NbB4, and (c) TaB4.Momentum-dependent superconducting gaps on the Fermi surface are shown for (d) VB4 at 6.0 K, (e) NbB4 at 5.0 K, (f) TaB4 at 6.0 K. Anisotropic superconducting gaps are shown for (g) VB4, (h) NbB4, and (i) TaB4 as a function of temperature.The variation of the superconducting gap (∆ k ) with temperature is obtained by solving the anisotropic Eliashberg equation with µ * = 0.1.(j) The vibration patterns of the phonon modes.The length of the arrows indicates the amplitude of atomic motions.
(g)).Secondly, the largest DOS at the Fermi level of NbB 4 is favorable for strong EPC and better superconductivity.Both the momentum-dependent superconducting gap on the Fermi surface and the temperature-dependent superconducting energy gap indicate that it has only one superconducting gap.And it shows the best superconductivity among the three structures, with the highest T c of 35.4 K (figure3(h)).Finally, figures 3(f) and (i) clearly show two superconducting gaps in TaB 4 .At 6.0 K, the two gaps vary from 2.23 meV to 3.08 meV and from 3.71 meV to 4.11 meV, respectively, and disappear at 18.6 K.
New J. Phys.25 (2023) 103019 Y Han et al 35.4 K. Our results inspire experimentalists to explore the fascinating superconducting properties of 2D borides and implement different layered structures to expand the family of 2D borides.