Introduction

The discovery of superconducting MgB21 with a critical temperature (Tc) of 39 K has ushered in a new era in the study of conventional high-temperature superconductors2. Metal borides (MB2) have since attracted considerable interest due to their potential for high Tc at ambient pressure. Transition-metal diborides, such as NbB23, ZrB24, and TaB25, have been experimentally identified and are characterized by a hexagonal crystal structure with P6/mmm space-group symmetry.

Following the successful fabrication of two-dimensional (2D) MgB2 (Tc = 36 K)6, attention has turned to 2D metal diborides. Recent density functional theory calculations have explored superconductivity in hexagonal layered transition metal diborides (MB2) for various metals (\(M={\rm{Sc}},{\rm{Zr}},{\rm{V}},{\rm{Nb}},{\rm{Ta}},{\rm{Cr}},{\rm{Re}}\)), yielding Tc values of 20.4, 2.9, 8.3, 35.5, 7.1, 4.5, and 2.4 K, respectively7. Additionally, research has extended to superconductivity in 2D metal tetraborides such as MB4 (M = Be, Mg, Ca, Sc, Al) with Tc values of 29.9, 22.2, 36.1, 10.4, and 30.9 K, respectively, as well as M2B2 (\(M={\rm{Mg}},{\rm{Re}}\)) with Tc values of 3.2 and 5.5 K, respectively.

For bulk conventional high-temperature superconductors, much research has been focused on hydrogen-rich materials (hydrides)8,9,10,11 since the prediction of superconductivity in metallic hydrogen12,13. While numerous calculations and experimental observations have been successful, their practical applications are limited by the requirement of extremely high pressures for stabilization. For instance, H3S has been experimentally observed to have Tc ~ 203 K at 150 GPa14,15,16, and LaH10 achieves Tc ~ 250 − 260 K at 170–180 GPa17,18,19,20,21,22.

To enable more practical applications, it is essential to explore hydrides that are stable at ambient pressure. Recently, the bulk semiconductor Mg(BH4)2 was shown to exhibit metallic behavior upon hole doping, achieving a Tc of up to 140 K under ambient conditions23. This has intensified interest in discovering hydrides that combine ambient pressure stability with high Tc. In addition, two-dimensional (2D) hydrides have emerged as promising candidates for high-temperature superconductivity.

2D hydrides can be classified into two categories. The first are obtained by replacing elements in 2D materials with hydrogen atoms, as seen in Janus transition-metal dichalcogenide hydrides (JTMDs)24,25,26,27,28,29,30,31,32. The second category involves hydrogenating 2D materials without removing their original atoms. Examples include hydrogenated graphene, achieving Tc > 90 K33; Mo2C3 with Tc = 53 K34; and CuH2 with Tc > 40 K35. 2D hydrogenated MgB2 has been predicted to exhibit Tc = 67 K, increasing to 100 K under 5% biaxial tensile strain36. Similarly, hydrogenated phosphorus carbide (HPC3) was predicted to exhibit Tc = 31 K, which increases to 57.3 K under 3% biaxial tensile strain37.

Metal borides structures are known to exhibit a high-frequency phonon spectrum due to the strong covalent bonding in boron vibrations. Upon hydrogenation, the addition of hydrogen atoms can further enhance the phonon density of states in the high-frequency region, potentially boosting phonon-mediated superconductivity. This dual contribution, from both boron and hydrogen, offers an advantage over other two-dimensional structures, where high-frequency phonons typically arise only from hydrogenation. Recent theoretical studies have also predicted superconductivity in 2D hydrogenated metal borides, M2B2H (M = Al, Mg, Mo, W), with Tc values of 52.64 K, 23.25 K, 21.54 K, and 18.67 K, respectively38. Furthermore, Ti2B2H4 was predicted to have Tc = 48.6 K, increasing to 69.4 K under 9% tensile strain39.

In this study, we systematically investigate 2D non-hydrogenated transition metal borides M2B2 (M = Sc, Y, V, and Nb), which have previously been suggested to be stable40,41, as well as novel mono-hydrogenated transition metal borides M2B2H and fully hydrogenated transition metal borides M2B2H4. We begin by discussing the lattice structure, stability, electronic structure, and then calculate their potential conventional high-temperature superconductivity.

Results

Crystal structures

The crystal structure of the M2B2H4 monolayer is illustrated in Fig. 1. Layered M2B2H4 adopts a 3D hexagonal structure with a space group symmetry of P6/mmm, analogous to Mg2B2. Figure 1 presents the side view and top view of M2B2Hx (x = 0,1,4) in panels (a) and (b), respectively. In this structure, boron atoms form a honeycomb lattice at the Wyckoff positions (1/3,2/3) and (2/3,1/2), sandwiched between two layers of transition metals. Hydrogen atoms at the Wyckoff positions (1/3,2/3) and (2/3,1/2) truncate the layers and passivate the surface.

Fig. 1: Crystal structures of M2B2H4 and M2B2H.
figure 1

a, b Shows side and top views of the 2D M2B2H4 structure (M = Sc, Y, V, and Nb) where the transition metals (M), boron (B), and hydrogen (H) atoms are represented by orange, green, and pink spheres, respectively. c, d Show side and top views of the 2D M2B2H with the wrinkle of the boron layers. Short “bonds” are show for clarity and have no chemical significance.

Full size image

In the case of mono-hydrogenated configurations, the hydrogen atom (H1) breaks symmetry and induces wrinkles in the honeycomb boron monolayer, as depicted in panels (c) and (d) of Fig. 1. The extent of wrinkling is quantified by the perpendicular distance between the two layers of wrinkled boron atoms, denoted as δB (B2:B1). This wrinkling only occurs when the hydrogenation is asymmetric, as in the cases of mono-, di-, and tri-hydrogenation. For the two-hydrogen case, we investigated three possible configurations: both hydrogen atoms on the top layer, and configurations where one hydrogen is on the top and the other on the bottom layer either aligned or offset (Supplementary Table. 1). Structure crystal structures of di- and tri-hydrogenation (Supplementary Fig. 1). Our calculated and optimized lattice parameters for a range of materials with normal metallic configuration are given in Table 1.

Table 1 The table shows the lattice constants (a), the distance between the two transition metal layers (M2:M1), the distance between boron atoms (B2:B1), the distance between the upper transition metal layer and the nearest upper-layer hydrogen atom (M2:H), and the vertical separation between the two wrinkled boron layers δB
Full size table

Compared to the non-hydrogenated cases, the effect of hydrogen is usually to slightly increases the interatomic spacings and lattice parameters.

To investigate possible magnetic ordering, we performed calculations for the non-magnetic state as well as magnetic configurations, including ferromagnetism and two types of antiferromagnetic order by introducing magnetic ordering on the transition metals. The two types of antiferromagnetic order differ by the opposite direction of magnetic moments between the two layers of transition metals. For all hydrogenated transition metal borides considered, we found that the non-magnetic metallic state is the most stable configuration. We also found that M2B2 compounds generally exhibit no magnetic behavior, with the exception of V2B2, which is antiferromagnetic.

The stability of these compounds was explored systematically. The phonon dispersion of M2B2 shows dynamical stable lattice dynamics. It was previously demonstrated40,41 that Sc2B2, Y2B2, V2B2, and Nb2B2 are dynamically and thermally stable. Similarly, hydrogenated M2B2Hx (x = 1,2,3,4) also have stable phonon. A detailed discussion of the phonon spectrum will be presented later.

In addition, we examined the energy associated with the formation of a crystal structure from its constituent elements, the cohesive energy given by

$${E}_{{\rm{coh}}}={E}_{2{\rm{D}}\text{-}{\rm{structure}}}-2{E}_{{\rm{M}}}-2{E}_{{\rm{B}}}-N{E}_{{\rm{H}}},$$
(1)

for M2B2H4, M2B2H and M2B2, respectively, where EM, EB and EH are isolated atoms of transition metals, boron and hydrogen atom, respectively. This is the relevant reference state when considering vapor deposition growth, which is the most likely way to synthesize under appropriate experimental conditions. Additionally, we evaluated the formation energy (Eform) of these 2D materials with respect to the most stable competing bulk phases42,43. These values, representing the energy required to form the crystal structure from its constituent elements in their standard states, are summarized in Table 2. The bulk phases of the transition metals are as follows: Scandium (Sc) and Yttrium (Y) crystallize in a hexagonal close-packed (hcp) structure with space group P63/mmc (No. 194); Vanadium (V) and Niobium (Nb) have a body-centered cubic (bcc) structure with space group Im\(\bar{3}{\rm{m}}\) (No. 229); Boron (B) exhibits a rhombohedral structure characteristic of α-boron with space group R\(\bar{3}{\rm{m}}\); and Hydrogen (H) exists as molecular hydrogen (H2) in its standard state. We also tested our analysis with bulk ScB2 and YB2. The enthalpies of formation were found to be −0.84 and −0.57 eV/atom, or equivalently −81.06 and −55.29 kJ/mol per atom, respectively. These values compare well with the theoretical results by Colinet and Tedenac44, which reported −80.99 and −54.59 kJ/mol, respectively, as well as with the experimental values of −102.3 and −35.7 kJ/mol. These results indicate that our analysis is sufficiently accurate and reliable.

Table 2 The table presents the cohesive energies and formation energies in units of eV per atom, with the values in parentheses given in kJ/mol per atom for various hydrogenation configurations
Full size table

Experimentally, the degree of hydrogenation can be tuned by varying the hydrogen gas pressure, temperature, and exposure time during synthesis. Hydrogen plasma treatment has been employed in recent studies to incrementally vary hydrogen content in 2D transition-metal materials45,46. As we shall see, although compounds of the form M2B2Hx (x = 1, 2, 3) can exhibit dynamical lattice stability due to positive phonons, they rarely exhibit superconductivity. Therefore, in this work, we focus on how mono-hydrogenation modifies the pristine structure and how full hydrogenation can give rise to high-temperature superconductivity, while di- and tri-hydrogenation, M2B2Hx with x = 2, 3, is not considered in detail; nevertheless, the phonon properties and Eliashberg spectral function of electron-phonon coupling are presented (Supplementary Fig. 2).

Electronic structures

To study the electronic properties of the 2D M2B2H4, M2B2H, and M2B2 compounds, we investigated their electronic structures, including the orbital-resolved electronic band structure, the electronic density of states (EDOS), the orbital-projected density of states (PDOS), and the Fermi surface as shown in Figs. 2 and 3. In general, M2B2H4, M2B2H, and M2B2 show metallic behavior, except for V2B2, which exhibits antiferromagnetic ordering. The electronic states at the Fermi level are generally dominated by the transition metal d-orbital electrons, which can be grouped into \({A}^{{\prime} }(d{z}^{2})\), \({E}^{{\prime} }({d}_{xy},{d}_{{x}^{2}-{y}^{2}})\), and E″(dyz, dxz) at the Γ point.

Fig. 2: Electronic structures of M2B2, M2B2H, and M2B2H4 (M = Sc, Y).
figure 2

The electronic structures of the 2D non-, mono-, and fully-hydrogenated compounds include the orbital-resolved band structure, the projected electronic density of states for the d-orbitals of the transition metals Sc and Y, the p-orbitals of boron atoms, the s-orbitals of hydrogen atoms, and the Fermi surfaces.

Full size image

The III-TM group (Sc, Y)

Without hydrogenation, Sc2B2 and Y2B2 exhibit similar electronic band structures. In Sc2B2, three bands span the Γ to K direction, a single band appears along the K to M direction, and an intersecting U-shaped band extends from M to Γ, crossing the Fermi level. This intersection forms a pocket of the Fermi surface, as shown in Fig. 2. A similar topology is observed for Y2B2, resulting in a very similar Fermi surface between Sc2B2 and Y2B2.

With mono-hydrogenation (M2B2H), the electronic structures are slightly modified. Several Fermi surface pockets merge into a single connected Fermi surface, as shown in Fig. 2 for Sc2B2H and Y2B2H. The electronic density of states remains largely unchanged, indicating that the hydrogen atom acts as a perturbation on the electronic bands due to the minimal overlap between the wavefunctions of the pristine system and the hydrogen atom.

For full hydrogenation (Sc2B2H4 and Y2B2H4), the electronic structures and density of states are significantly altered, as shown in Fig. 2. The EDOS at the Fermi level decreases substantially due to the bonding of d-orbital electrons with the additional hydrogen atoms. For Y2B2H4, the d-orbitals no longer dominate the Fermi surface entirely. The PDOS of the px, py, and pz orbitals of boron and the s orbital of hydrogen become comparable to the d-orbital contributions from Y atoms. In contrast, for Sc2B2H4, the d-orbitals still dominate. Near the Γ point, flat bands close to the Fermi level result in a high electronic density of states characteristic of a van Hove singularity (vHs).

The V-TM group (V, Nb)

For Nb, a similar scenario to (Sc, Y) is observed, where the electronic bands and the electronic density of states (EDOS) of Nb2B2H are only slightly perturbed by the addition of a single hydrogen atom to the pristine compound Nb2B2. For Nb2B2H, the d-orbital-dominated bands shift slightly due to hydrogenation, but the Fermi surface topology remains largely unchanged compared to the pristine Nb2B2. The exception is the open two-shell Fermi surfaces around the M point, which become a single open Fermi surface due to the absence of a crossing band along the M to Γ direction, as depicted in Fig. 3. Despite favoring antiferromagnetic ordering, we show metallic electronic properties of V2B2, as shown in Fig. 3, as a reference. The anti-ferromagnetic electronic structure is presented (Supplementary Fig. 3). Upon mono-hydrogenation, it undergoes an electronic phase transition from an antiferromagnetic to a normal metallic phase.

Fig. 3: Electronic structures of M2B2, M2B2H, and M2B2H4 (M = V, Nb).
figure 3

The electronic structures of the 2D non-, mono-, and fully-hydrogenated compounds include the orbital-resolved band structure, the projected electronic density of states for the d-orbitals of the transition metals V and Nb, the p-orbitals of boron atoms, the s-orbitals of hydrogen atoms, and the Fermi surfaces.

Full size image

Under fully hydrogenation, unique characteristics emerge in the electronic band structure at the Fermi level. For V2B2H4, a flat dispersion band is observed near the M point along the Γ to M direction and close to the Γ point, resulting in a high EDOS with vHs at the Fermi level, as shown in Fig. 2. In contrast, for Nb2B2H4, although peaks of vHs are present, the Fermi level does not coincide with any of these singularities.

A key distinction between the V-TM (V, Nb) and III-TM (Sc, Y) groups is the behavior under fully hydrogenation. For the III-TM group, the EDOS decreases significantly with increased hydrogenation, while for the V-TM group, it follows the opposite trend. In Sc2B2H4 and Y2B2H4, there is a significant contribution from the B p-orbitals and H s-orbitals, as illustrated in the projected electronic density of states in Fig. 2. In particular, the feature near the K point likely arises from the substantial contribution of B pz orbitals, as shown in the orbital-resolved band structure. This suggests that V-TM (V, Nb) and III-TM (Sc, Y) groups may exhibit different electron-phonon coupling and superconductivity, a topic that will be discussed later.

These results indicate that the evolution of electronic band topology and Fermi surfaces can be slightly modified or heavily disrupted depending on the extent of additional compositional elements. This suggests a tunability of electronic properties in these systems, perhaps by mixing the TMs.

Phonon properties

Figure 4 shows the phonon spectrum of hydrogenated transition metal borides of M2B2H4, M2B2H, and M2B2. There are no imaginary modes, so all structures are dynamically stable.

Fig. 4: Phonon properties and electron-phonon coupling of M2B2-based monolayers.
figure 4

These plots show the weighted electron-phonon coupling (EPC) dispersion of the phonon, the projected phonon density of the states (PHDOS), the isotropic Eliashberg spectral function α2F(ω), and the spectrum-dependent electron-phonon coupling λ(ω) of 2D M2B2H4, M2B2H, and M2B2.

Full size image

Pristine materials

The labeling of the modes 1-12 in Table 3, is arranged according to their energy in from Sc2B2. For ease of comparison, similar eigenmodes in other materials are given the same label. Therefore, the ordering of the energy spectrum need not follow the labeling Table 3. This is also similar to Tables 4 and 5 for M2B2H and M2B2H4, respectively.

Table 3 Table shows bands, subgroups, and eigenvectors of M2B2 (M = Sc, Y, Nb) where the direction of vibration referred as out-of-plane (OP) or in-plane (IP)
Full size table
Table 4 Table shows bands, subgroups and eigenvectors of M2B2H (M = Sc, Y, V, Nb) where the direction of vibration referred as out-of-plane (OP) or in-plane (IP)
Full size table
Table 5 Table shows bands, subgroups and eigenvectors of M2B2H4 (M = Sc, Y, V, Nb), where the direction of vibration referred as out-of-plane (OP) or in-plane (IP)
Full size table

For M2B2, the three acoustic modes, namely the longitudinal in-plane acoustic mode (LA), the transverse in-plane acoustic mode (TA), and the flexural out-of-plane acoustic mode (ZA), range from higher to lower energy, respectively, as shown in Fig. 4. The nine optical modes as shown in Fig. 5a consist of two degenerate in-plane vibrations (modes 4 and 5) and an out-of-plane vibration (mode 6) of the transition metal (TM). The other optical modes are mainly associated with boron motion, comprising an out-of-plane vibration (mode 9) and in-plane vibrations (modes 11 and 12). Sc2B2 and Y2B2 are similar to each other, except for the effects of the larger mass of yttrium. Table 3 shows eigenvalues of energy spectrum at the Γ point, where the eigenvectors for each mode of M2B2 are depicted in (a) of Fig. 5. V2B2 and Nb2B2 also exhibit a similar phonon dispersion, aside from the shift due to the TM mass effect. In the case of M2B2, we can identify that the most energetic phonon mode is in-plane boron vibration (mode 11,12).

Fig. 5: The figures show the optical vibrational eigenvector modes and their corresponding symmetry subgroups for Sc2B2 (modes 4-12) and Sc2B2H (modes 4-15).
figure 5

For other transition metals (Y, V, Nb), the mode order may differ, as shown in Tables 3 and Table 4.

Full size image

Mono-hydrogenated materials

The lightly hydrogenated M2B2H has fifteen eigenvectors at the Γ point, with three additional modes compared to their original systems, as shown in Fig. 5(b). The three additional high-frequency optical modes (modes 13, 14, and 15) are well-separated from the other nine optical modes. These modes arise primarily from in-plane hydrogen vibrations (modes 13 and 14) and an out-of-plane hydrogen vibration (mode 15).

For Nb2B2H, the eigenvectors and eigenvalues of energy spectrum are similar to those of the Sc2B2H and Y2B2H compounds, as listed in Table 4. The eigenvalues of energy spectrum are ordered with respect to the vibrations of Sc2B2H, as shown in Fig. 5b. However, in the case of V2B2H, Table 4 shows a differently ordered energy spectrum with respect to the eigenvectors of Sc2B2H, as illustrated in Fig. 5b. Away from the Γ point, the contribution of boron vibrations becomes significant in the high-energy spectrum of V2B2H, as shown in the projected phonon density of states in Fig. 4. In the case of M2B2H, we can identify that the most energetic phonon mode is out-of-plane hydrogen vibration (mode 15).

Fully hydrogenated materials

The fully hydrogenated M2B2H4 introduces an additional nine modes, significantly modifying the optical modes, as shown in the phonon projected density of states in Fig. 4.

For Sc2B2H4 and Y2B2H4, the optical phonon spectrum can be classified into two regions: a middle energy range (~40 meV to 100 meV) and a high energy range (above 140 meV), as shown in Fig. 4. In the middle phonon energy range, the spectrum arises from boron and hydrogen vibrations, as illustrated in the phonon projected density of states in Fig. 4. At the Γ point, this energy spectrum corresponds to optical phonon modes ranging from modes 7 to 16 within the energy range of 40 meV to 100 meV, as shown in Table 5, ordered with respect to the eigenvectors of Sc2B2H4 as shown in Fig. 6.

Fig. 6: Eigenvectors of selected phonon modes in Sc2B2H4.
figure 6

The figures show the optical vibrational eigenvector modes and their corresponding symmetry subgroups of Sc2B2H4 (modes 4–24), where other transition metals (Y, V, Nb) the order could be different according Table 5.

Full size image

Above 100 meV, no phonons are observed until 140 meV, where a high-energy phonon spectrum appears. At the Γ point, this high-energy spectrum corresponds to optical phonon modes ranging from modes 17 to 24 within the energy range of 140 meV to 160 meV. These modes primarily originate from hydrogen vibrations, as indicated in the phonon projected density of states in Fig. 4. In Sc2B2H4 and Y2B2H4, the most energetic phonon modes are in-plane hydrogen vibrations (modes 23 and 24).

For V2B2H4 and Nb2B2H4, there is no gap in the phonon spectrum, with contributions from both boron and hydrogen vibrations throughout the spectrum. The significant difference between V2B2H4 and Nb2B2H4 lies in the most energetic phonon modes. For Nb2B2H4, the highest-energy modes correspond to in-plane hydrogen vibrations (modes 21 and 22), whereas for V2B2H4, the most energetic modes are in-plane boron vibrations (modes 13 and 14), as shown in Sc2B2H4-eigenvectors ordered Table 5. This distinction is also evident in the phonon-projected density of states in Fig. 4.

When comparing V2B2H4 and Nb2B2H4 with Sc2B2H4 and Y2B2H4, a notable difference is observed in the energies of out-of-plane and in-plane hydrogen vibrations. The out-of-plane hydrogen vibrations in Nb2B2H4 and V2B2H4 exhibit higher energies than those in Sc2B2H4 and Y2B2H4. However, the in-plane hydrogen vibrations in Nb2B2H4 and V2B2H4 have lower energies than those in Sc2B2H4 and Y2B2H4. The eigenvector-ordered phonon spectrum at the Γ point for M2B2H4 is summarized in Table 5.

Bonding and phonon properties

The significantly high phonon frequencies originating from boron vibrations observed in V2B2H4 can be attributed to the strengthening of B–B covalent interactions. This interpretation is supported by both structural data and Crystal Orbital Hamilton Population (COHP) analysis, where we report the integrated value up to the Fermi level as the Integrated Crystal Orbital Hamilton Population (ICOHP). Specifically, the calculated B–B bond lengths increase from the non- to the fully hydrogenated phases as shown in Table 1 in the scandium (1.80 Å to 1.84 Å) and yttrium (1.90 Å to 1.97 Å) systems. In contrast, the bond lengths decrease in the vanadium (1.70 Å to 1.69 Å) and niobium (1.79 Å to 1.76 Å) systems over the same hydrogenation range. These structural trends indicate stronger B–B bonding in the vanadium- and niobium-based systems. This observation is further supported by the COHP results, as shown in Table 6, which confirm stronger B–B interactions in the vanadium-based compound. Therefore, V2B2H4 consistently exhibits the strongest B–B bonding across different hydrogenation levels, resulting in the highest-energy in-plane boron phonon modes, which would also explain different maximum boron spectrum across M2B2H (M = Al, Mg, Mo, W) in ref. 38.

Table 6 Table shows ICOHP values in the unit of eV for B-B and TM-H (transition metal-hydrogen) bonds in M2B2H4 (M=Sc, Y, V, Nb)
Full size table

The significantly low phonon frequencies originating from hydrogen vibrations observed in V2B2H4 can be attributed to the weakening of hydrogen bonding, primarily involving the transition metal (TM). This interpretation is supported by both structural data and crystal orbital Hamilton population (COHP) analysis. Specifically, the calculated TM–H layer thickness decreases in the scandium (1.06 Å to 0.78 Å) and yttrium (1.17 Å to 0.82 Å) systems as the hydrogen content increases from mono- to full hydrogenation. In contrast, the TM–H layer thickness increases in the vanadium (0.96 Å to 0.98 Å) and niobium (1.06 Å to 1.08 Å) systems under the same conditions. These structural trends indicate stronger TM–H bonding in the scandium- and yttrium-based systems, which correlates with their relatively high-energy hydrogen phonon modes. This conclusion is further supported by COHP results, as shown in Table 6, which confirm the weakest TM–H bonding in the vanadium-based system. Therefore, V2B2H4 consistently exhibits the weakest TM–H interactions, resulting in the lowest-energy hydrogen phonon modes.

Thus, both structural and electronic bonding analyses support the conclusion that the transition-metal-dependent evolution of bonding strength is the primary origin of the variations observed in the boron and hydrogen vibrational spectra. Specifically, the strongest bonding between boron atoms leads to the highest frequency of in-plane B vibrations, whereas the weakest bonding between vanadium and hydrogen results in the lowest frequency of in-plane H vibrations among the four transition metals.

Phonon-mediated superconductivity

The electron-phonon coupling (EPC) is investigated through the weighted EPC dispersion of the phonons and contour plots of EPC (λqv), as illustrated in Figs. 4 and 7, respectively. In general, the EPC mainly comes from near the Γ point.

Fig. 7: Figures show comparison of superconducting properties for Sc2B2H4, V2B2H4, and Nb2B2H4.
figure 7

Top row: Momentum-resolved electron-phonon coupling strength λqν plotted across the Brillouin zone for phonon branches ν = 1–24. Middle row: Temperature dependence of the superconducting gap(s). Sc2B2H4 exhibits two distinct superconducting gaps, while V2B2H4, and Nb2B2H4 show a single-gap behavior. Bottom row: Fermi surface distribution of the superconducting gap Δ(k) at 10.0 K with energy window of 0.025 eV, illustrating anisotropy and multi-gap features. The color scale represents the magnitude of Δ(k) in meV.

Full size image

For Sc2B2H4, the main contribution arises from the optical vibration modes ν = 7−10, specifically the out-of-plane hydrogen vibrations (ν = 7, 8) and the out-of-plane boron vibrations (ν = 9, 10). In the case of V2B2H4, the EPC is predominantly from vibration modes ν = 6−8 and ν = 15−16. Additionally, contributions are observed from various vibration modes away from the Γ point. This behavior is evident in the weighted EPC dispersion of the phonon in the energy range around 70 meV, particularly between the K point and the M point, as shown in Fig. 4. This leads to a high average electron-phonon coupling function of the Eliashberg function, with significant contributions around 30 meV, 70 meV, and 100 meV, resulting in a large λ value of 1.29. For Nb2B2H4, the Eliashberg function exhibits two distinct regions: a lower-energy region and a higher-energy region. At lower energies (around 20 meV), the EPC arises from niobium phonon excitations away from the Γ point. At higher energies, the EPC is distributed across the Brillouin zone and is dominated by the optical vibrations of boron and hydrogen. Consequently, the net contour map of EPC reveals peaks of λqv throughout the Brillouin zone, resulting in the highest value of λ among the compounds studied here.

According to Eq. (6), the Eliashberg spectral function depends on the product of the transition matrix, the Fermi nesting function, and the phonon spectrum. To obtain a high superconducting temperature (Tc), we need large values of electron-phonon coupling (λ) and a high logarithmic average of the phonon spectrum \({\omega }_{\log }\). Calculated values are shown in Table 7 together with other possible superconductors that have previously been studied.

Table 7 Quantities important for superconducting calculations, including the electron-phonon coupling constant λ, the logarithmic average of the phonon energy \({\omega }_{\log }\), the critical superconducting temperature T\({}_{c}^{AD}\) based on the Allen-Dynes formula, and the critical superconducting temperature T\({}_{c}^{AD}\) based on the closing of the superconducting gap
Full size table

Non-hydrogenated M2B2, are not superconductors (predicted as millikelvin Tc, which we neglect here), and mono-hydrogenation barely changes their superconducting properties. For all monohydrides, we notice only minor changes in λ \({\omega }_{\log }\). Therefore, mono- hydrogenation does not induce superconductivity, except weakly for V2B2H which could be superconducting at about 1.3K. For full hydrogenation of the Scandium and Yttrium compounds, we still have a low electron-phonon coupling (λ), but we obtain a significant increase in \({\omega }_{\log }\). This leads to a higher of Tc = 18.7 K, in Sc2B2H4 and Tc = 2.74 K in Y2B2H4. The V-TM group has significant changes in both λ and \({\omega }_{\log }\) due to full hydrogenation. These high values of λ lead to impressively high predicted Tc, with some dependence on whether the Allen-Dynes formula or the closing of the superconducting gap is used to calculated Tc. We find V2B2H4, 53 K up to 83 K, and for Nb2B2H4, at least 54 K up to 69 K.

Our results show that Sc2B2H4 indeed exhibits two superconducting gaps, as evidenced by the non-vanishing gap values ranging from ~10 meV to 6 meV, as shown in the superconducting gap plot at 10.0 K. We apologize for the earlier oversight, which has now been corrected. This observation is further supported by the contour plot of the superconducting gap, which clearly reveals two distinct gaps located near the Γ and K symmetry points. The first superconducting gap, ranging from 10 meV to 6 meV, appears near the Γ point, while the second gap, centered around the K point, exhibits lower energies concentrated near 4 meV. In Sc2B2H4, there is a significant contribution from the B p-orbitals and H s-orbitals, as illustrated in Fig. 2. The superconducting gap near the K point likely arises from the substantial contribution of B pz orbitals, as shown in the orbital-resolved band structure in Fig. 2.

In contrast, the gap near Γ is primarily associated with d-orbital contributions. This behavior differs from that observed in V2B2H4 and Nb2B2H4, where there is negligible contribution from the B p- and H s-orbitals. As a result, only a single superconducting gap is observed in these materials, which originates solely from the d-orbitals.

High-Tc ambient 2D-BCS materials

Possible high-Tc superconductors have been intensively investigated in the form of 2D materials. Since the discovery of a high Tc in bulk MgB2, many studies have focused on exploring potential high-Tc BCS superconductors based on the well-established BCS theory and Migdal-Eliashberg theory. These investigations primarily aim to design materials that achieve high values of electron-phonon coupling (λ) and average logarithmic frequency (\({\omega }_{\log }\)).

In our study, we identify two materials with potential high superconducting transition temperatures based on the Allen-Dynes formula and Migdal-Eliashberg equations. Specifically, we find Tc values ranging from 53 K to 83 K with high values of electron-phonon coupling (λ) of 1.29 and 1.34 for V2B2H4 and 54 K to 69 K for Nb2B2H4. Within the same famility, Ti2B2H4 has also been studied with Tc of 48 K. These 2D materials demonstrate the potential as high-Tc superconductors among other high-Tc 2D-BCS superconductors.

For bulk MgB2 with a Tc of 39 K1, a monolayer counterpart of MgB2 has been predicted with Tc of 20 K47 and experimentally shown to be superconducting at 36 K of superconducting MgB2 thin film on graphene6, which is close to the Tc of bulk MgB2. Moreover, hydrogenated h-MgB2 has also been shown to exhibit a Tc boosted to 67 K36. Additionally, a similar compound, Mg2B4C2, which belongs to the BCS family of MgB2, has been predicted to have a Tc of 46 K48. The monolayer LiCB has also been predicted to be high-Tc BCS 2D superconductors with Tc of 70 K49. The hydrogenated h-LiCB has also been reported with Tc of 80 K50. We summarized these high-Tc BCS superconductors as shown in Table 8.

Table 8 Table shows electron-phonon coupling coupling λ, the logarithmic average of the phonon energy \({\omega }_{\log }\), the square average of the phonon energy ω2, and the critical superconducting temperature Tc of listed high-Tc 2D-BCS superconductors at ambient
Full size table

Discussion

In summary, we have performed a systematic first-principles study of two-dimensional transition metal borides M2B2 (M = Sc, Y, V, Nb) under varying degrees of hydrogenation. All mono-hydrogenated (M2B2H), moderately hydrogenated (M2B2H2,3), and fully hydrogenated (M2B2H4) phases are dynamically stable and exhibit metallic ground states, with negative cohesive and formation energies. Regarding the effects of hydrogenation, mono-hydrogenation induces only minor changes in the electronic structure, whereas full hydrogenation significantly alters the band topology and enhances both electron-phonon interactions and phonon-mediated superconductivity.

In the III-transition metal (III-TM) group (Sc, Y), which has fewer valence electrons, the electronic density of states at the Fermi level is relatively low, with comparable contributions from d-orbitals of the transition metal, p-orbitals of boron, and s-orbitals of hydrogen. In contrast, the V-transition metal (V-TM) group (V, Nb), which has more d-valence electrons, exhibits a higher density of states, particularly in V2B2H4, where a pronounced van Hove singularity appears near the Fermi level. This elevated electronic density of states in the V-TM compounds promotes stronger electron-phonon coupling, ultimately favoring higher superconducting transition temperatures.

Another key contrast emerges between the III-TM and V-TM compounds. In V2B2H4 and Nb2B2H4, enhanced B–B covalency, evidenced by shorter bond lengths and stronger electron localization, leads to stiffened boron phonon modes. This enhancement arises from greater electron donation by V and Nb, owing to their higher valence electron count, which more effectively populates B–B bonding states than in the Sc and Y systems. Simultaneously, the weaker and more delocalized metal–hydrogen bonds in the V/Nb compounds suppress hydrogen-related vibrational frequencies, in contrast to the stronger and more localized M–H interactions observed in the Sc/Y counterparts. As a result, the combination of low-frequency hydrogen modes and high-frequency boron modes in the V-TM compounds generates a significant phonon density of states over a broad spectral range.

Together with the high electronic density of states, this enhances the electron-phonon coupling constant (λ), leading to markedly higher superconducting critical temperatures. Notably, V2B2H4 and Nb2B2H4 exhibit Tc values of up to 83 K and 69 K, respectively, underscoring their potential as high-Tc two-dimensional superconductors.

Finally, our analysis shows that Sc2B2H4 exhibits two superconducting gaps near the Γ and K points. The additional gap near the K point arises primarily from B pz orbitals, while the gap near the Γ point is dominated by d orbitals. In contrast, V2B2H4 and Nb2B2H4 display a single gap mostly derived from d-orbital contributions, reflecting the negligible involvement of B p and H s orbitals. These findings highlight the critical role of orbital composition in shaping the superconducting behavior.

Methods

Electronic structure calculations

All calculations are based on Density Functional Theory (DFT), as implemented in the well-established QUANTUM ESPRESSO (QE) package51,52. Therefore, in this investigation, we employ standard DFT. Norm-conserving pseudopotentials53,54 and the Perdew-Burke-Ernzerhof (GGA-PBE)55 are used for the exchange-correlation energy functional with wave function and charge density cutoffs of 80 Ry and 320 Ry, respectively. The vacuum thickness was set to be 30 Å to make sure no overlap of the wavefunction between monolayers. The crystal structures were fully relaxed with a force threshold of 1.0−5 eV/Å. The crystal structure was visualized using VESTA56 with layered hexagonal transition metal with P6/mmm space-group symmetry. For electronic structure calculations, we used 24 × 24 × 1 k-point grid for self-consistent calculations to sample the reciprocal space of the Brillouin zone. The electronic density of states and Fermi surfaces were computed using the optimized tetrahedral method for the calculation of non-self-consistency,57, and the Fermi surface was visualized by XCRYSDEN58. In addition, we studied several magnetic phases, including ferromagnetism (FM), antiferromagnetism, specifically A-type (AAF) and G-type (GAF) configurations, using 2 × 2 × 1 supercells with a 12 × 12 × 1 k-point grid.

Electron-phonon coupling calculations

To obtain the isotropic Eliashberg spectral function for M2B2 and M2B2H, we computed the interatomic force constants (IFC) by performing the Fourier transform of the matrix elements \({g}_{{\boldsymbol{k}}+{\boldsymbol{q}},{\boldsymbol{k}}}^{{\boldsymbol{q}}\nu ,mn}\) of the coarse 12 × 12 × 1 q-mesh grid calculated by using Density Functional Perturbation Theory (DFPT)59 implemented in QE. From corrected IFCs, the phonon linewidth, γqν, was computed,

$${\gamma }_{{\boldsymbol{q}}\nu }=2\pi {\omega }_{{\boldsymbol{q}}\nu }\sum _{nm}\sum _{{\boldsymbol{k}}}{\left\vert {g}_{{\boldsymbol{k}}+{\boldsymbol{q}},{\boldsymbol{k}}}^{{\boldsymbol{q}}\nu ,mn}\right\vert }^{2}\delta ({\epsilon }_{{\boldsymbol{k}}+{\boldsymbol{q}},m}-{\epsilon }_{F})\delta ({\epsilon }_{{\boldsymbol{k}},n}-{\epsilon }_{F})$$
(2)

and the electron-phonon coupling, λqν, associated with the phonon wavevector q and the phonon mode of μ, as

$${\lambda }_{{\boldsymbol{q}}\nu }=\frac{{\gamma }_{{\boldsymbol{q}}\nu }}{\pi N({\epsilon }_{{\bf{F}}}){\omega }_{{\boldsymbol{q}}\nu }^{2}}.$$
(3)

The Eliashberg spectral function, α2F(ω), is obtained by

$${\alpha }^{2}F(\omega )=\frac{1}{2N({\epsilon }_{F})}\sum _{{\boldsymbol{q}}\nu }\frac{{\gamma }_{{\boldsymbol{q}}\nu }}{{\omega }_{{\boldsymbol{q}}\nu }}\delta (\omega -{\omega }_{{\boldsymbol{q}}\nu }).$$
(4)

To investigate M2B2H4, we employ the electron-phonon Wannier-Fourier interpolation method60,61,62 within the EPW package63,64 to accurately compute the superconducting properties, specifically the electron-phonon coupling (λ) and critical temperature (Tc). This methodology also allows for the detailed analysis of the anisotropic Migdal-Eliashberg theory65,66,67,68,69,70 by solving the two coupled nonlinear anisotropic Migdal-Eliashberg equations,

$${Z}_{nk}(i{\omega }_{j})=1+\frac{\pi T}{N({\varepsilon }_{F})}\sum _{m{k}^{{\prime} }{j}^{{\prime} }}\frac{{\omega }_{{j}^{{\prime} }}}{\sqrt{{\omega }_{{j}^{{\prime} }}^{2}+{\Delta }_{m{k}^{{\prime} }}^{2}(i{\omega }_{{j}^{{\prime} }})}},$$
(5)
$${Z}_{nk}(i{\omega }_{j}){\Delta }_{mk}(i{\omega }_{j})=\frac{\pi T}{N({\varepsilon }_{F})}\sum _{m{k}^{{\prime} }{j}^{{\prime} }}\frac{{\Delta }_{m{k}^{{\prime} }}(i{\omega }_{{j}^{{\prime} }})}{\sqrt{{\omega }_{{j}^{{\prime} }}^{2}+{\Delta }_{m{k}^{{\prime} }}^{2}(i{\omega }_{{j}^{{\prime} }})}}\left[\lambda (nk,m{k}^{{\prime} },{\omega }_{j}-{\omega }_{{j}^{{\prime} }})-{\mu }^{* }\right]\delta ({\epsilon }_{m{k}^{{\prime} }}-{\varepsilon }_{F}),$$
(6)

self-consistently along the imaginary axis at the fermion Matsubara frequencies ωj = (2j + 1)πT. For these calculations, we use the QUANTUM ESPRESSO package with the same computational settings as previously mentioned, followed by Wannier-Fourier interpolation of maximally localized Wannier functions (MLWF) (Supplementary Fig. 4) onto k- and q-point grids of 200 × 200 × 1 and 100 × 100 × 1, respectively. Employing dense grids ensures the convergence of λ values, as indicated by the stability of the corresponding α2F(ω) and λ(ω) even as the k- and q-point grid densities increase. The Fermi surface thickness was set to 0.55 eV, with a Matsubara frequency cutoff at 1.35 eV. Dirac δ functions were broadened using a Gaussian function with widths of 0.1 eV for electrons and 0.5 meV for phonons when analysing V2B2H4, and N2B2H4. Due to maximum phonon frequency, the numerical parameters for Sc2B2H4 were set accordingly. The Fermi surface thickness was set to 0.68 eV, with a Matsubara frequency cutoff at 1.7 eV, while we used the same Dirac δ functions for electrons and for phonons. Lastly, the Morel-Anderson pseudopotential was set to μ* = 0.1 for practical purposes.

The superconducting transition temperature (TC) was determined using the semi-empirical Allen-Dynes formula71:

$${T}_{c}={f}_{1}{f}_{2}\frac{{\omega }_{\log }}{1.20}\exp \left(-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{* }(1+0.62\lambda )}\right)$$
(7)

where the electron-phonon coupling constant λ is derived from the Eliashberg spectral function:

$$\lambda =2\mathop{\int}\nolimits_{0}^{\omega }d\Omega \left(\frac{{\alpha }^{2}F(\Omega )}{\Omega }\right),$$
(8)

and the logarithmic average phonon frequency is calculated as:

$${\omega }_{\log }=\exp \left(\frac{2}{\lambda }\mathop{\int}\nolimits_{0}^{\infty }d\Omega \log (\Omega )\left(\frac{{\alpha }^{2}F(\Omega )}{\Omega }\right)\right).$$
(9)

The correction factors f1 and f2 are given by:

$${f}_{1}{f}_{2}={\left(1+\left(\frac{\lambda }{2.46(1+3.8{\mu }^{* })}\right)\right)}^{1/3}\times \left(1+\frac{{\lambda }^{2}\left(\frac{{\omega }_{2}}{{\omega }_{\log }}-1\right)}{{\lambda }^{2}+3.31{(1+6.3{\mu }^{* })}^{2}}\right).$$
(10)

This f1f2 correction factor is applied when the electron-phonon coupling constant λ exceeds 1.0. The mean-square phonon frequency (ω2) is given by

$${\omega }_{2}=\sqrt{\frac{2}{\lambda }\mathop{\int}\nolimits_{0}^{{\omega }_{\max }}{\alpha }^{2}F(\omega )\omega d\omega }.$$
(11)

Although it would be interesting to perform electron-phonon coupling calculations using DFT + U, where the Hubbard parameter U is used to localize the d-orbitals, this approach is not fully supported for linear-response calculations in the current publicly available version of QUANTUM ESPRESSO. However, it would be an issue for future investigation.

Crystal orbital hamilton population (COHP) calculations

To investigate the bond strength in the studied 2D materials, the COHP analysis was performed using the LOBSTER package72,73,74.