Introduction

The discovery of highly crystalline two-dimensional (2D) superconductors1,2,3,4,5, such as NbSe2 monolayer6,7,8,9, has provided possibilities for van der Waals (vdW) heterostructures nanoengineering of insulator-superconductor interfaces10 and 2D Josephson junctions, without the need of an insulating layer11. One main challenging issue in the realization of 2D superconductivity is that most of the well-known conventional bulk superconductors either do not superconduct or poorly superconduct when their dimensions are reduced6,7,8,12,13,14,15,16,17. Although numerous 2D phonon-mediated superconductors have recently been predicted from first-principles calculations, the highest predicted intrinsic Tc stayed around 20 K17,18,19,20,21,22,23,24 (19 K for B2C monolayer21, 10.3 K for B2O monolayer17, and 19–25 K for borophenes22, to name a few). Though in some cases Tc has been enhanced by means of the chemical doping, intercalation, strain, and/or substrate proximity effects17,23,25,26,27,28,29,30,31,32, it is essential to discover intrinsic 2D superconductors that exhibit high-Tc without any doping or tuning of external parameters (here high-Tc does not refer to unconventional superconductivity as in case of cuprates or iron-based superconductors33).

Among all the Bardeen–Cooper–Schrieffer (BCS) type conventional superconductors, MgB2 stands out with a record Tc of 39 K, the highest reported Tc at zero-pressure34,35,36. Such a high-Tc in MgB2 stems from the strong electron-phonon (el-ph) coupling occurring primarily due to the in-plane stretching of B-B bonds (i.e., E2g phonon modes), which strongly couple with the self-doped charge carriers from magnesium to boron atoms26,35,36,37,38. Remarkably, only two (E2g) out of a total of nine phonon modes contribute strongly to the total el-ph coupling in MgB235,36,37,38,39,40,41,42,43,44. Once the fundamental mechanism of such a high-Tc in bulk MgB2 was understood, which by the way was a subject of intense research for over a decade period26,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, researchers started proposing ways to augment Tc through rational materials design approach25,37,43,44,50,51,52. Pickett and co-workers proposed that one can, in principle, achieve a much higher Tc (than 39 K) by designing a MgB2-like stable material that has a similar Fermi surface as in MgB2, and in which more than two phonon modes couple to the electronic states near the Fermi level, thereby, resulting in a sizable total el-ph coupling25,43,44. This idea has been employed for the rational design of bulk superconductors with a good success rate53,54,55,56,57,58,59,60,61,62,63,64. The high-pressure superconductivity observed at 250 K in lanthanum hydride is one such example65,66,67,68.

Despite the large success with the bulk conventional superconductors, two-dimensional intrinsic superconductors having a high-Tc remained elusive. Notably, various attempts have been made to realize superconductivity in the 2D analogues of bulk MgB213,32,51,69,70,71,72,73,74. On the one hand, Xu and Beckman proposed a quasi-2D MgB2 nanosheet with inert surfaces, which turns out to be a semiconductor with a bandgap of 0.51 eV resulting from the quantum confinement effects13. On the other hand, Bekaert et al. reported that a considerably high-Tc of 20 K can be realized in monolayer MgB2 without surface passivation, i.e., if only such a material with a highly chemically reactive surface could be made73,74. In a recent study, Bekaert et al. theoretically demonstrated that a MgB2 monolayer can be stabilized by adding hydrogen adatoms. Interestingly, they find that the hydrogenation process leads to a high-Tc of 67 K, which can be further boosted to over 100 K by means of a biaxial strain on the hydrogenated MgB2 monolayer32. While an experimental validation of the predicted Tc in monolayer MgB2 is still missing, the aforementioned theoretical works markedly enhance our understanding of superconductivity in 2D materials.

In this work, we present a MgB2-like 2D material – Mg2B4C2, having charge neutral inert surfaces, which is predicted to superconduct at a strikingly high-Tc in the 47–48 K range (predicted using the McMillian-Allen-Dynes theory75,76,77), which is among the highest Tc yet reported for an intrinsic 2D material without any doping, strain or substrate-induced effects. The main advantageous feature in 2D Mg2B4C2 is the fact that, unlike in bulk MgB2, more than two phonon modes strongly couple to the electronic states near the Fermi level, thus, resulting in a substantially larger el-ph coupling (λ = 1.40) in monolayer Mg2B4C2 compared with the bulk MgB2 (λbulk = 0.7338, and 0.6136). We note that the estimated λ in monolayer Mg2B4C2 is comparable to the predicted λ = 1.46) in hydrogenated MgB2 monolayer32. Moreover, our calculations reveal nontrivial topological electronic features in Mg2B4C2 exhibiting Dirac cones and practically gapless Dirac nodal lines at the Fermi level near the corner points of the hexagonal Brillouin zone (BZ), which enhance the density of states (DOS) at the Fermi level by almost 30% compared to that of bulk MgB2, hence, positively contributing towards a higher Tc.

Results and discussion

Material design strategy

We start by describing our rationale for design of a stable MgB2-like 2D superconductor having inert surfaces. Generally, layered vdW materials can be exfoliated to produce their 2D analogues78. Although bulk MgB2 has a layered structure, it is not a vdW material. Bulk MgB2 crystallizes in space group P6/mmm (#191) containing alternating layers of Mg and B atoms stacked along the \(\overrightarrow{c}\) lattice direction, as shown in Fig. 1(a)42. The bonding between the Mg and B atoms is purely ionic, which means that Mg atoms donate two electrons to B atoms, thereby making each Mg 2+ and each B 1−. Since a B− is isoelectronic to a charge-neutral carbon atom, a B-B sheet is structurally analogous to a single layer graphene, but it has a different ordering of bands than those of graphene. A simple exfoliation of MgB2 into a 2D slab with B (or Mg) termination would yield a highly reactive electron-rich (or hole-rich) surface layer that is chemically unstable.

Fig. 1: Crystal structure.
figure 1

a Bulk MgB2, and (b) side (as viewed from \(\overrightarrow{{{{\bf{a}}}}}\)) and top views of monolayer Mg2B4C2 (Mg: orange, B: green, C: black). Solid black lines mark the unit cell boundaries, and shaded grey areas represent vacuum in the left panel of (b). The region marked by dashed black lines in (b) can be arbitrarily repeated (see text).

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We propose that one can passivate the charged surface layers in the MgB2 slab by systematically substituting one boron by one carbon atom at the top and bottom surfaces of the slab. Figure 1(b) shows the top and side views of a Mg2B4C2 monolayer designed using the aforementioned strategy. Strikingly, we find that modestly repeating the intermediate Mg-B layers, i. e. , the layers sandwiched between the top and bottom surfaces (highlighted using dashed rectangle in Fig. 1(b)), thereby making thicker slabs of (MgB2)nC2 while remaining in the quasi-2D limit, n being the total number of Mg layers, retains the key features of the Mg2B4C2 monolayer. The electronic bandstructures calculated up to n = 5 are shown in the Supplementary Fig. 1. This feature could be particularly useful in the experimental realization of 2D superconductivity in Mg2B4C2. We note that MgB2 monolayer can also be passivated by an appropriate hydrogenation process32.

Mg2B4C2 monolayer, shown in Fig. 1(b), belongs to the layer group \(p\bar{3}m1\) (#72) having DFT (PBE) optimized lattice parameters a = b = 2.87 Å. The absolute thickness between the top and bottom atomic layers is 7.14 Å, whereas, the interlayer spacing between the adjacent Mg and B-B, and Mg and B-C (C-B) layers is ~1.8 Å, and ~1.7 Å, respectively. We note that the inversion symmetry is preserved due to the inverted ordering of the top and bottom layers in the structure shown in Fig. 1(b). However, one could break the inversion symmetry by replicating the top and bottom layers, i.e., by making the top and bottom layers alike, either both as B-C or both as C-B. Our calculations suggest that the structure with inversion symmetry is energetically more favorable (5 meV/f.u.) than the structure with broken inversion symmetry; although both structures are dynamically, elastically, and mechanically stable since they exhibit all positive phonon frequencies, positive elastic constants, and satisfy the Born-Huang mechanical stability criteria (see Supplementary Table 2 and Supplementary Fig. 6). The only qualitative difference in the electronic properties of the structure with broken inversion symmetry is a small lifting of some band degeneracies at the K high symmetry point (see Supplementary Fig. 3). This effect is analogous to the application of a perpendicular electric field to a bilayer graphene79.

In this article, hereafter, we focus only on ground state structure of a monolayer Mg2B4C2 with preserved inversion symmetry. We note, all other possible atomic configurations of this composition are higher in energy. Furthermore, our exfoliation energy calculations (see Supplementary Table 3) suggest that the reported monolayer Mg2B4C2 belongs to the “easily exfoliable" category, as classified by Mounet et al.80.

Topological electronic properties of Mg2B4C2 monolayer

After describing the crystal structure and its stability, we now focus on the topological electronic properties of Mg2B4C2 monolayer. We begin by summarizing the key features of the electronic structure of bulk MgB242 from which Mg2B4C2 monolayer is derived. As shown in Fig. 2(a), the Fermi surface of MgB2 is composed of boron p orbitals, where px,y orbitals hybridize with s orbitals to form strong covalent in-plane σ bonds at the zone center, while the unhybridized pz orbitals form relatively weak out-of-plane π bonds at zone boundaries (Mg acts as electron donor). Due to such a distinct Fermi-surface geometry, two superconducting gaps exists in bulk MgB2: (i) the stronger σ gap of ~7 meV, and (ii) the weaker π gap of ~2–3 meV35,36,39,46,81,82,83,84,85. Different symmetries of the σ and π bonds largely suppress the impurity scattering in MgB239,41,42,85.

Fig. 2: Electronic structure.
figure 2

Atomic orbitals projected electronic band structure of (a) bulk MgB2, and (b) monolayer Mg2B4C2 calculated without spin-orbit coupling (SOC) along the high symmetry direction of BZ. Cyan, red, and blue colors represent the contribution from the s, px,y, and pz orbitals, respectively. See Supplementary Fig. 4 for more details. c Calculated Fermi surface of monolayer Mg2B4C2. Light pink/green, and grey colors depict hole/electron, and intertwined electron-hole pockets, respectively. d Energy bandgap (Egap) plotted in color scale (eV units) in the vicinity of a K high-symmetry point. The dashed circle marks the k-loop along which Berry phase was computed. e The local electronic density of states of the (100) and (010) edge states spectrum. Red/White color denotes the states near the edge/interior of the 2D system. Topological nontrivial edge states are marked using arrows.

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Since the basic structure and charge neutrality of MgB2 is preserved in monolayer Mg2B4C2, the electronic band structure of monolayer Mg2B4C2 qualitatively resembles with that of the bulk MgB2, as shown in Fig. 2(a, b), but with some additional features. For instance, there is a set of degenerate σ bands (σouter) present at Γ below the Fermi level arising from the px,y orbitals of the outer boron-carbon layers. The other set of degenerate σ bands (σinner) at Γ that cross the Fermi level (also present in MgB2) are formed by the px,y orbitals of the inner boron-boron layer. These two sets of σ bands are almost parallel and split by ~1.6 eV at Γ. Since the σouter bands are completely occupied, they should, in principle, have no contribution in superconductivity, unless there is a large external field applied in a FET-like geometry86.

In addition to the set of σ bands at Γ, we notice the presence of Dirac-like band crossings at the K point, as well as along the high-symmetry directions near the K point of monolayer Mg2B4C2. Regardless of their topological nature, these band crossings at the Fermi level, highlighted using a dashed magenta box in Fig. 2(b), yield a large DOS at the Fermi level (almost 30% larger than in bulk MgB2), which contributes substantially to the total el-ph coupling in the studied monolayer. We note that the Dirac-like crossing at K is also present in bulk MgB2, but it is situated well-above the Fermi level87. The Dirac-like band crossings in Mg2B4C2 monolayer are formed by highly dispersing pz orbitals of carbon and boron atoms (see Supplementary Fig. 4). Thus, the Fermi surface of Mg2B4C2 monolayer, shown in Fig. 2(c), embodies three main features: (i) two-hole pockets at Γ (one circular and another that takes the shape of the BZ) composed of σ bonded boron px,y orbitals, (ii) an electron pocket at M formed by boron pz orbitals, and (iii) intertwined electron and hole pockets at the K point and along K–M high-symmetry line, formed by π bonded carbon and boron pz orbitals. We note that all these pockets show very strong coupling to the phonon modes, and, as a result, they play the key role in governing superconductivity in Mg2B4C2 monolayer, as we discuss later. Furthermore, the sharp and well-defined (almost flat) boundaries of the charge-carrier pockets at the Fermi surface set up the stage for the possible realization of Kohn-like divergencies88, and charge-density-wave ordering89,90 in this 2D system, which is beyond the scope of present work and calls for a more comprehensive attention in the future.

By plotting the energy bandgap (Egap) distribution in the vicinity of the K points, we discover presence of a triangular nodal line in the vicinity of each K point, as shown in Fig. 2(d). However, this is not a truly gapless nodal line since a small Egap (~5 meV) exists due to the subtle breaking of Mz mirror symmetry. It is worth noting that the Dirac point at K is protected by the C3v rotation, inversion, and time-reversal symmetries; a small gap opens at Dirac points when the inversion symmetry is broken by making the top and bottom B-C layers identical91. Although there are theoretical proposals suggesting the possibility of topological superconductivity in Dirac semimetals92, we think that the so-far studied models are quite simple, and this topic requires a more thorough examination before any exotic effects can be confidently claimed here.

In order to prove the nontrivial topological nature of Dirac points, we compute the Berry phase along a k-loop enclosing the gapless point at K, as marked using dashed lines in Fig. 2(d). Our calculations yield a nontrivial Berry phase of π for Dirac points at K. We note, this exercise could not be performed for the Dirac nodal line near K because enclosing the nodal line residing in the kx-ky plane would require a k-loop encircling along kz and kz is not defined for a 2D system. Nevertheless, the presence of time-reversal and spatial-inversion symmetries of Mg2B4C2 monolayer enables us to determine the Z2 topological invariants using the Fu-Kane criterion93. The inversion parity eigenvalues of the electronic wavefunction of all 12 occupied bands at four time-reversal invariant momenta (TRIM) points are given in Table 1. The product of all parity eigenvalues (δ) at each TRIM is also listed in Table 1. We find that the Z2 topological index is nontrivial due to δ = −1 at three TRIM points. Here, we note that bulk MgB2 has a weak Z2 topological index (0; 001) due to the band-inversions occurring at the Γ and A (0, 0, 0.5) high-symmetry points of 3D hexagonal BZ87. Robust topological surface states have recently been experimentally observed in bulk MgB294.

Table 1 Parity eigenvalues of all occupied bands and their products at four TRIM points.
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Since the nontrivial topology in 2D systems is often manifested in the gapless 1D edge states, we further confirm the nontrivial topological features of monolayer Mg2B4C2 by computing the local density of states at (100) and (010) edges of 60 unit cell thick nano-ribbons. Topologically nontrivial 1D edge states connecting band-crossing points were obtained at both (100) and (010) edges, as shown in Fig. 2(e), thus, proving the nontrivial topology of the Mg2B4C2 monolayer.

Electron-phonon coupling and superconductivity in Mg2B4C2

We find that the roots of superconductivity in Mg2B4C2 monolayer are the same as in bulk MgB235,36,37,38,39,40,41,42,43,44. However, the main advantageous factor in Mg2B4C2 is that, in addition to the doubly degenerate E2g modes that govern superconductivity in MgB2, numerous other phonon modes strongly couple to the electronic states near the Fermi level yielding a much larger overall el-ph coupling, and thus, resulting in a considerably higher Tc.

The calculated phonon spectrum of Mg2B4C2 monolayer, shown in Fig. 3(a), contains a total of 24 phonon modes (8 atoms/cell) having the following mode symmetry at Γ:

$$\begin{array}{l}{{{\Gamma }}}_{{{{\rm{acoustic}}}}}={A}_{2{{{\rm{u}}}}}\oplus {E}_{{{{\rm{u}}}}},\,\,{{\mbox{and}}}\,\\ {{{\Gamma }}}_{{{{\rm{optic}}}}}=4{A}_{1{{{\rm{g}}}}}\oplus 3{A}_{2{{{\rm{u}}}}}\oplus 3{E}_{{{{\rm{u}}}}}\oplus 4{E}_{{{{\rm{g}}}}}.\end{array}$$
(1)

Here, A1g and Eg are Raman-active modes, whereas, A2u and Eu are infrared-active modes. In Fig. 3(c–f), we show the atomic vibration patterns for the four phonon modes, namely, three nondegenerate A1g modes (indices 14, 18, and 19) and one degenerate Eg mode (indices 16-17), which exhibit the dominant el-ph coupling. All these A1g modes correspond to the out-of-plane vibrations of the Mg, inner B-B, and outer B-C layers, while the Eg mode corresponds to the in-plane stretching of the inner B-B layer. The A1g modes primarily modulate the el-ph coupling associated with the π bonded pz orbitals contributing to the electron and hole pockets located at the BZ boundaries. Whereas, the doubly degenerate Eg mode couples with the σ bonded px,y orbitals forming the hole pockets located at Γ. Here, it is worth noting that the higher frequency Eg modes (indices 21-22) that correspond to the in-plane stretching of the outer B-C layers do not make a significant contribution to the overall el-ph in this system, which is as expected since these modes modulate the occupied σouter bands located well-below the Fermi level at Γ [see Fig. 2(b)]. However, these modes may participate in superconductivity when the system is doped with p-type charge carriers32.

Fig. 3: Electron-phonon coupling.
figure 3

a Calculated phonon spectrum of Mg2B4C2 monolayer with phonon linewidths λ(q, n) plotted using shaded blue color. To avoid large overlap of λ(q, n) with the phonon spectra, we have divided the intensity by a factor of two. The colored circles mark the three out-of-plane nondegenerate A1g modes (indices 14, 18, and 19), and the yellow diamond marks one in-plane doubly degenerate Eg mode at Γ. These modes exhibit dominant el-ph coupling. The atomic displacement patterns corresponding to these modes are shown in (c–f). The nondegenerate A1u mode (index 15) marked using symbol ` × ' does not contribute to the total el-ph coupling, although it appears to be buried in the large λ(q, n) of the Eg mode. The numerals 14, 15, 16, 17, 18, and 19 denote the phonon mode index as counted from the lowest to the highest frequency modes (i.e.,1-3 for acoustic modes). Mg atoms are omitted in (c) for the sake of clarity. The Eliashberg spectral function α2F(ω) along with the el-ph coupling constant λ in plotted in the right panel of (a). b Estimated Tc as a function of the μ* parameter.

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Since the electronic and vibrational band structures of inner B-B and outer B-C layers are essentially independent of each other, we predicate that the reported properties of the studied Mg2B4C2 monolayer would be retained even when the number of the inner B-B layers are repeated (until a critical thickness), thus making the system thicker. This feature might greatly simplify the eventual realization of superconductivity in Mg2B4C2.

To quantify the superconducting properties of Mg2B4C2 monolayer, we employ the McMillian-Allen-Dynes theory derived from the isotropic Migdal-Eliashberg formalism75,76,77 which relies on the calculation of the el-ph coupling matrix elements within DFT. The calculated matrix elements correspond to the transition probabilities of different Kohn-Sham states induced by a change in the potential due to a small ionic displacement. Thus, these matrix elements provide the main ingredients to calculate the el-ph coupling strength and the Eliashberg spectral function α2F(ω) as a function of the phonon frequency ω. Since the physical process behind the phonon-mediated superconductivity is the exchange of a phonon between two electrons, a strong el-ph coupling is desired to achieve a high-Tc in a BCS superconductor. Theoretical details of such calculations are explained in numerous other papers82,95,96.

In Fig. 3(a), we plot the calculated phonon linewidth λ(q, n) for each phonon mode n at each wave vector q using blue color. Note that the plotted phonon linewidth is scaled down by a factor of two to avoid large overlap with the neighboring phonon branches. The largest contribution to the total el-ph coupling strength comes from three nondegenerate A1g modes and one doubly degenerate Eg mode, as marked in Fig. 3(a). We note that the A2u mode (index 15), marked using `×' in Fig. 3(a), does not contribute to the total el-ph coupling, although it appears buried in the large λ(q, n) overlap from the Eg mode. Notably, in addition to the aforementioned A1g and Eg phonon modes, various other modes make relatively smaller contributions to the overall el-ph coupling strength, as revealed by the Eliashberg spectral function α2F(ω) plot shown in the right panel of Fig. 3(a).

In addition to the el-ph coupling, the net phonon linewidth λ(q, n) can have some contribution from the phonon-phonon (ph-ph) interactions owing to the phonon anharmonicity36. Therefore, we thoroughly investigate ph-ph interactions by computing ph-ph linewidth using the ab-initio molecular dynamics simulations. In this approach, we mapped the forces, obtained from the finite-temperature molecular dynamics simulations, evaluated in a 3 × 3 × 1 supercell onto a model Hamiltonian describing the lattice dynamics. This temperature dependent effective potential (TDEP) technique97,98 enabled us to calculate the third-order response from the effective renormalized interatomic force constants. Our calculations revealed that the ph-ph linewidths are an order of magnitude smaller than the el-ph linewidths. The maximum value of obtained ph-ph linewidth is ~2 meV, which is much smaller compared to the el-ph linewidth values that are typically larger than ~ 70 meV in the studied system. This result implies that, although the system inherits some anharmonic effects, we can safely discard the ph-ph contributions in the study of its superconducting properties.

Based on the BCS theory of superconductivity and above results, we estimate the critical temperature Tc using the McMillian-Allen-Dynes formula99,100,101:

$${T}_{c}=\frac{{\omega }_{log}}{1.2}exp\left[-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{* }(1+0.62\lambda )}\right],$$
(2)

where ωlog is the logarithmic averaged phonon frequency, λ is the total el-ph coupling constant, and μ* is the effective screened Coulomb repulsion constant with a typical value ranging from 0.04 to 0.16 (see Table 2)21,35,36,38,56. We obtain λ by integrating the cumulative frequency-dependent el-ph coupling λ(ω) given by the following expression:

$$\lambda (\omega )=2\int\nolimits_{0}^{\omega }\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega$$
(3)

We find a fairly large value of λ = 1.40, which is considerably larger than the one reported for bulk MgB2 (λbulk = 0.7338, and 0.6136). We observe that the estimated Tc does not vary drastically as a function of μ*, as shown in Fig. 3(b). This is consistent with an earlier work by Choi et al.36, which reported that the superconducting properties of MgB2 are not very sensitive to the μ* parameter within the isotropic McMillian-Allen-Dynes formalism. We note that for bulk MgB2, μ* = 0.05 has been used to get the correct estimate of Tc ~ 40 K38. Therefore, using the McMillian-Allen-Dynes formula99,100,101, we estimate the Tc of Mg2B4C2 monolayer to be in the range 47–48 K without any doping or strain. Our results are consistent with a recent study32 in which Tc = 67 K and λ = 1.46 was predicted in a hydrogenated MgB2 monolayer by solving the fully anisotropic Eliasberg equations. We argue that the predicted Tc in Mg2B4C2 monolayer can be further enhanced by biaxial strain17,32 or by p-doping32. In passing, we would like to mention that the predicted Tc could moderately vary if a fully anisotropic Migdal-Eliashberg theory32,36,82,102 or SC-DFT103,104,105,106 is employed. This is particularly important here because the applicability of the McMillian-Allen-Dynes formula becomes limited in the case of large el-ph coupling.

Table 2 Listing of superconducting parameters required for the prediction of Tc using the McMillian-Allen-Dynes formula for some reported 2D phonon-mediated superconductors (data for bulk MgB2 are included for comparison). This table includes data of effective Coulomb screening parameter μ*, electronic DOS at the Fermi level N(EF) (in states/spin/Ry/cell), logarithmic averaged phonon frequency ωlog (in K), total electron-phonon coupling constant λ, and estimated Tc (in K). Experimental Tc values are noted in the table (see the Supplementary Table 1 for more details).
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In order to highlight the novelty of our results, in Table 2 we list the theoretical superconducting parameters along with the estimated Tc for some reported 2D phonon-mediated superconductors (see Supplementary Table 1). The good agreement between the experimental data for LiC6102,107,108, 2H-NbSe27,109,110,111, and C6CaC6112,113 and the corresponding theoretical results obtained from the McMillian-Allen-Dynes theory validate the the predictive power of the McMillian-Allen-Dynes theory.

Summary

In summary, we present a 2D material Mg2B4C2, similar to MgB2, but with inert surfaces obtained by the replacement of outer B-B layers by B-C layers. Our calculations suggest that this structure is dynamically, elastically and mechanically stable. It also features a nontrivial topological electronic band structure together with a large el-ph coupling (λ = 1.40), which is more than twice as large as that of the bulk MgB2 and comparable to that of in a hydrogenated monolayer MgB232. Use of the standard McMillian-Allen-Dynes theory predicts the superconducting transition temperature Tc to be in the range of 47–48 K without any doping or tuning of external parameters such as strain. To the best of our knowledge, this is among the highest predicted intrinsic Tc in a conventional BCS-type 2D superconductor. In addition to the large el-ph coupling, the presence of sharp and well-defined flat boundaries of the charge-carrier pockets at the Fermi surface imply the possible realization of Kohn-like divergencies and charge-density-wave ordering in this 2D system, which calls for a dedicated study in future.

Methods

DFT calculations

The electronic bands structure and phonon calculations were performed using density-functional theory (DFT) as implemented in the VASP114,115,116,117. The phonopy118 and PyProcar119 tools were used for the post-processing of data. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional120 and PAW pseudo-potentials121,122 were used. The employed k-point grid for self-consistent calculations was 30 × 30 × 1, and the cutoff for the kinetic energy of plane waves was set to 700 eV. A vacuum of thickness ~ 30 Å was added to avoid the periodic interactions along the c-axis. Since the spin-orbit coupling (SOC) effects were found to be negligible in the studied system, SOC was not included in the reported calculations. The elastic and mechanical properties were analyzed using the MechElastic code123,124. The exfoliation energy was calculated using four different exchange-correlation approximations: the (PBE) GGA approximation120, the SCAN125 meta-GGA, vdW-DF2 GGA functional126, and SCAN together with the rVV10 correlation functional (SCAN+rVV10)127. The topological properties of Mg2B4C2 were studied by fitting the DFT calculated bandstructure to a real space tight-binding Hamiltonian obtained using the maximally localized Wannier functions (MLWFs) approach128,129. The local density of states at (100) and (010) edges were calculated for 60 unit cells thick nano-ribbons using the WannierTools package129 with vacuum added along the c-axis of the ribbon.

For the electron-phonon coupling matrix elements calculations, we used the abinit package130,131,132,133,134. We employed norm conserving pseudopotentials (using the ONCVPSP scheme of Hamann135), and a plane wave basis set up to kinetic energies of 35 Ha. Cell parameters were optimized by using the PBE exchange-correlation functional as in VASP calculations. We used a uniform grid of 18 × 18 × 1 for the ground state calculations, and a phonon grid of 9 × 9 × 1 for the phonon part. A total of 288 el-ph matrix elements were calculated. Calculations of the phonon interatomic force constants, and the el-ph coupling matrix elements performed in this work used the second-order perturbation theory136,137. The temperature dependent effective potential (TDEP) technique97,98 was used to study the phonon-phonon interactions and phonon anharmonic effects.