2D materials have attracted considerable focus after the preparation of the graphene [1], which shows the excellent thermal and catalytic performances [2–4]. While the zero bandgap limits the application of graphene in power devices [5, 6], and then the transition metal dichalcogenides (TMDs) materials are proposed with decent bandgaps larger than that of the bulk one [7]. For example, the MoS₂ monolayer presents novel optical absorption characteristics as a potential photocatalyst [8, 9], which also can be prepared as photoluminescence [10]. In particular, the Janus MoSSe monolayer, as popular asymmetric TMDs, further demonstrates a novel thermal and phononic properties with a polar nature [11–13]. Likewise, 2D Janus materials explain different characteristic on both sides, such as adsorbed [14], catalytic [15], mechanical [16] and electronic properties [17]. All these obtained novel performances of the 2D materials also can be tuned by strain engineering [18, 19], interface coupling [20, 21], external electric field [22] and temperature [23] etc.

Using the nanoscale materials as a catalyst in the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) is also popular [24–26], because more active sites can be exposed [27–30]. For example, the ability of the OER of the CoO₂ and FeO₂ monolayers can be improved by decreasing 40% overpotential under external strain [31]. The barrier of the biphenylene network in HER is obtained as low as −0.03 eV by the decent atomic doping [32]. The intrinsic defect is also a popular strategy to tune the HER and OER performances of the 2D materials [33]. Besides, to further extend the application range of 2D material, the heterostructure is constructed, which can induce novel electronic and catalytic properties because of the built-in electric field across the interface [17]. PtS₂/arsenene heterostructure is constructed with a −0.487 eV potential for the HER, which is lower than the origin PtS₂ and arsenene monolayers [34]. C₂N/WS₂ heterostructure can facilitates OER with potential of about 1.81 eV [35]. Besides, the prediction of new 2D materials is also an important approach to expand the properties for nano-devices [36, 37]. For example, Ag₂S monolayer acts a semiconductor with auxetic mechanical properties using as nano-electronics [38]. The band edge positions of the SnP₂S₆ monolayer promises the redox potential of the water splitting as a photocatalyst [39]. IV–VI monolayers present ultrahigh carrier mobility, which also act as an excellent HER catalyst [40]. Recently, 2D boron based compound has been proposed to possess excellent electronic and catalytic properties. For example, the Janus B₂P₆ is predicted as potential photocatalyst for water splitting [41], and the band edge energy also can be tuned by external strain [42]. The Li₂B₂ monolayer is calculated with a high hole mobility of 6.8 × 10³ cm²⋅V^{− 1}⋅s^{− 1} using as high-speed electronic devices [43]. The auxetic B₄N monolayer shows an apparent mechanical anisotropy coupled with robust structural stability in future nano-mechanical devices [44]. All these point that exploring the boron (B) based 2D materials as advanced functional material presents significant prospects.

In this work, we propose a novel 2D materials, X₂B₆ (X = K, Na and Rb) monolayer, using elemental mutation method by the prototype of Li₂B₆ monolayer [43]. The stability of the predicted K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is addressed by phonon spectrum and the ab initio molecular dynamics (AIMD) simulations. The mechanical and the electronic performances are investigated by the density functional theory (DFT). Then, the ultra-high electron mobility and the optical light absorption properties are obtained in the Na₂B₆ monolayer. The unique catalytic activity of these X₂B₆ monolayer in HER and OER is studied.

All the calculations in this investigation were implemented by Vienna ab initio simulation package (VASP) using first-principle method, which is based on the DFT [45–47]. The core electrons was addressed in the simulations using projector augmented wave potentials (PAW) [48, 49]. The Perdew–Burke–Ernzerhof (PBE) was carried out to demonstrate the exchange correlation method based on the generalized gradient approximation [50–52]. To correct the weak van der Waals interaction in the HER and OER system, the DFT-D3 method was used by Grimme functional [53]. Furthermore, the Heyd–Scuseria–Ernzerhof (HSE06) calculations were explored to investigate the electronic and optical performances of the Na₂B₆ monolayer [54]. It is worth noting that the spin effect was not taken into account in the calculation of electronic properties, because we found that the obtained band structure with the spin turned on and off are exactly the same, shown as Supplementary Figure S1. The Monkhorst–Pack with a k-point grids as 11 × 11 × 1 and 17 × 17 × 1 were used in the relaxation and self-consistent simulations, respectively. The vacuum space was set as 25 Å, which can optimize the interaction of nearby layers. The parameter of the convergence for force and energy are set as 0.01 eVÅ⁻¹ and 0.01 meV, respectively. In the simulation of the phonon spectra, the PHONOPY code was used based on the density functional perturbation theory [55, 56].

First, the atomic structure of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are predicted with the puckered crystal structure showing a space group of Pca ₂₁ , by elemental mutation method using the prototype of structure from the Li₂B₆ monolayer [43]. The optimized structure of the X₂B₆ monolayer is presented as Figure 1A and the obtained lattice parameters of the a (or b) in unit-cell of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are 4.311 (or 3.554), 4.313 (or 3.616) Å and 4.312 (or 3.552) Å, respectively, which is smaller than that of the Li₂B₆ monolayer. The bond length of X–B (L _XB) and the B–B (L _BB) in the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are obtained as L _XB = 2.850 Å and L _BB = 5.018 Å, L _XB = 2.523 Å and L _BB = 3.598 Å, L _XB = 2.978 Å and L _BB = 5.450 Å, respectively. Furthermore, the cohesive energy of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is calculated as 6.794 eV/atom, 5.974 eV/atom and 6.048 eV/atom, respectively, which is obtained by (2E _X + 6E _B–E _XB)/8, where E _X, E _B and E _XB are used to present the total energy of an X, B atoms and the X₂B₆ system, respectively. Thus, the calculated cohesive energy of the X₂B₆ system is also larger than that the predicted IV–VI system (about 3.37–3.81 eV/atom) [57] and comparable with the CB monolayer (about 6.13 eV/atom) [58], showing a stability for these K₂B₆, Na₂B₆ and Rb₂B₆ monolayers. Besides, the dynamic stability of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is also studied by phonon spectra obtained in Figure 1B. Obviously, no imaginary frequency can be found in the phonon spectra of these X₂B₆ monolayer, suggesting the dynamic stability of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers. The highest frequency of the optical branch of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is about 34 THz which is smaller than the prototype of the Li₂B₆ system.

Then, the thermal stability of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is also investigated by the AIMD method by the Nosé−Hoover heat bath functional [59]. The supercell of the X₂B₆ monolayer is obtained as 7 × 4 × 1 to prevent the lattice translational constraints, which also presents 192 atoms [60]. Besides, the structure of the X₂B₆ monolayer is totally relaxed under 300 K and 600 K for 10 ps, after the complete calculations. One can see that the crystal structure of the X₂B₆ monolayer is still undamaged shown as the insets of Figure 2. The temperature and energy of the X₂B₆ monolayer system in the AIMD calculations are also convergent demonstrated as Figure 2, explaining a clear thermal stability at 300 K. Furthermore, the K₂B₆ and Na₂B₆ monolayers are also stable at 600 K because the structure is still intact, while the structure of the Rb₂B₆ monolayer can be melted down at 600 K, shown as Figure 2.

Then, the mechanical properties of these X₂B₆ monolayer is investigated by the orientation dependences of Young’s modulus using [61] Equation 1 as follows: E θ = C 11 C 22 − C 12 2 C 11 ⁡ sin 4 ⁡ θ + C 22 ⁡ cos 4 ⁡ θ + C 11 C 22 − C 12 2 C 66 − 2 C 12 cos 2 ⁡ θ ⁡ sin 2 ⁡ θ (1) where θ explains the angle of a direction shown as Figure 1A. The calculated Young’s modulus of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is demonstrated in Figures 3A, C, E, respectively. One can see that K₂B₆, Na₂B₆ and Rb₂B₆ monolayers present the anisotropic Young’s modulus with the maximal and minimal values at θ = 90° and θ = 0°, respectively, shown as Figures 3A, C, E. The obtained maximal Young’s modulus of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are 296 N/m, 397 N/m and 406 N/m, respectively. Then, the orientation dependent Poisson’s ratio of the X₂B₆ monolayer is also studied by [57] Equation 2 as follows: v θ = − C 11 + C 22 − C 11 C 22 − C 12 2 C 66 cos 2 ⁡ θ ⁡ sin 2 ⁡ θ − C 12 cos 4 ⁡ θ + sin 4 ⁡ θ C 11 ⁡ sin 4 ⁡ θ + C 22 ⁡ cos 4 ⁡ θ + C 11 C 22 − C 12 2 C 66 − 2 C 12 cos 2 ⁡ θ ⁡ sin 2 ⁡ θ (2)

The calculated Poisson’s ratio of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are demonstrated by Figures 3B, D, F, respectively. Obviously, the maximal Poisson’s ratio of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers are obtained as about 0.35, 0.29 and 0.26, respectively, with the θ about 45°. Such obtained Young’s modulus and Poisson’s ratio of the X₂B₆ monolayer is also higher than that of the carbon monochalcogenides [62] and biphenylene [61].

Furthermore, the band structure of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is calculated shown by Figure 4 using PBE method. The K₂B₆ and Rb₂B₆ monolayers present a semi-metallic property, shown as Figures 4A, C, while the Na₂B₆ monolayer suggests semiconductor nature with the direct bandgap that the conduction band minimum (CBM) and the valence band maximum (VBM) are located at the Г point, shown as Figure 4B. In order to obtain a more accurate bandgap of the Na₂B₆ monolayer, HSE06 functional is explored. Interestingly, the Na₂B₆ monolayer presents an ultra-narrow bandgap as about 0.42 eV, smaller than the As₂X₃ system [63], shown as Supplementary Figure S2. It is worth noting that such ultra-narrow bandgap in Na₂B₆ monolayer is also reported in the PbN/CdO heterostructure (about 0.128 eV) [64], which can serve as a promising efficient nano-electronic and catalyst [65, 66]. Besides, the projected band structure of the Na₂B₆ monolayer is also demonstrated by Figure 4B. Obviously, B atoms almost contribute to the band energy comparing with the Na atoms.

Since the ultra-narrow bandgap is obtained as about 0.42 eV for the Na₂B₆ monolayer, the potential application as nano-devices is promising. Thus, the carrier mobility of the Na₂B₆ monolayer is necessary to be investigated. The electrons and holes mobility of the Na₂B₆ monolayer along the transport directions (a and b demonstrated in Figure 1A) is explored using the Bardeen-Shockley method [45] which is calculated by Equation 3 as follows: μ = e ℏ 3 C / k B T m * m x * m y * E 2 (3) where elementary charge, the Planck constant and the Boltzmann constant are e, ћ and k _B, respectively. The effective mass of the electron and hole is represented using the m*, which is calculated by Equation 4 as follows: m * = ℏ 2 d 2 E k d k 2 − 1 (4) where k and E _k are the wave vector and electronic energy, respectively. C represents the elastic modulus of the monolayered Na₂B₆, which is obtained by C = [∂² E/∂((l–l ₀ )/l ₀ )/S ₀. In this equation, the original lattice constant, the free energy and difference of the lattice constant by the strain are expressed as l, E and l ₀ , respectively. S ₀ is used to represent the area of the Na₂B₆ monolayer. The energy difference of the Na₂B₆ system by the external uniaxial strain is calculated as Figure 5A. Furthermore, E is the potential constant of the Na₂B₆, which is obtained using E = ΔE _edge/((l–l ₀ )/l ₀), where the ΔE _edge is difference of the CBM or VBM energy tuned by external strain in the a or b directions. As shown in Figure 5B, the CBM and the VBM of the Na₂B₆ monolayer can be obviously increased and decreased, respectively, when the external strain is applied. Besides, Figure 5B demonstrates that the dependence of VBM energy of the Na₂B₆ monolayer is obvious under applied strain, suggesting the large potential constant in the Na₂B₆ monolayer for holes.

Next, the calculated effective mass of the Na₂B₆ monolayer along a and b directions are shown as Table 1. One can see that the effective mass of electrons and holes is relatively uniform in transport direction. The calculated deformation potential constant of the hole is larger than that of the electrons in the Na₂B₆ monolayer shown as Table 1. Besides, the elastic modulus of Na₂B₆ monolayer is also explained as Table 1. It is worth noting that the elastic modulus of the Na₂B₆ monolayer is obtained as 409 N.m⁻¹ and 420 N.m⁻¹, respectively, which is consistent with the previous calculation results of Young’s modulus along a and b directions. Therefore, the apparent isotropic carrier mobility of the Na₂B₆ monolayer is also obtained that electron shows a fast mobility as about 9942 cm².V⁻¹.s⁻¹ and 5486 cm².V⁻¹.s⁻¹ along a and b directions, respectively. While the hole mobility in Na₂B₆ monolayer is calculated as 650 cm².V⁻¹.s⁻¹ and 862 cm² V⁻¹ s⁻¹, along a and b directions, respectively. In the same transport direction, the huge difference between electrons and holes allows them to be effectively separated, about 15 (a direction) times and 6 (b direction) times, suggesting the potential application as photocatalyst. Besides, the calculated electron mobility of the Na₂B₆ monolayer is even higher than that of other 2D materials, such as B₂P₆ monolayer (5888 cm².V⁻¹.s⁻¹) [42], Li₂B₆ monolayer (6800 cm².V⁻¹.s⁻¹) [43] and MoSi₂N₄ (2169 cm².V⁻¹.s⁻¹) [67].

Considering the ultra-narrow bandgap obtained for semiconductor of the Na₂B₆ monolayer, the optical absorption spectrum is further calculated by HSE06 method, which is defined as [8] Equation 5 as follows: α ω = 2 ω c ε 1 2 ω + ε 2 2 ω 1 / 2 − ε 1 ω 1 / 2 (5) where ε ₁(ω) shows the real parts and the ε ₂(ω) suggests the imaginary part of the dielectric constant. ω is demonstrating the angular frequency. While the complex dielectric function is calculated by ε(ω) = ε ₁(ω) + iε ₂(ω), where ε ₁ can be calculated from ε ₂ via the Kramers–Kronig relation. Furthermore, the ε ₁(ω) and ε ₂(ω) can be decided as Equation 6 as follows: ε 2 q → O u ^ , ℏ ω = 2 e 2 π Ω ε 0 ∑ k , v , c ∣ ⟨ Ψ k c u ^ ⋅ r Ψ k v ⟩ 2 × δ E k c − E k v − E (6) where Ψ k , E k and u ^ are the wave function, energy and unit vector of the electric field of the incident light. The superscripts (v and c) in Ψ k , E k , label the conduction bands and valence bands, respectively.

Shown as Figure 6A, the optical absorption ability is presented that the light absorption peak of the Na2B6 monolayer is about 11.8 × 105 cm−1 with the wavelength about 335 nm. Such excellent optical absorption performance of the Na2B6 monolayer is more advantages than that of other 2D materials such as AlN/Zr2CO2 heterostructure (3.79 × 105 cm−1) [68], CdO/Arsenene heterostructure (8.47 × 104 cm−1) [69] and SiSe monolayer (7.98 × 105 cm−1) [57].

Then, the catalytic properties of K2B6, Na2B6 and Rb2B6 monolayers are investigated by calculating the Gibb’s free energy of the system. First, the overall process of HER and the OER in water splitting is demonstrated as Equations 7–10 as follows: H 2 O + 2 h + →   2 H + + 1 / 2 O 2 (7) 2 H + + 2 e −   →   H 2 (8) where the main reactions in the HER process are: * + H + + e –   →   H * (9) H * + H + + e –   →   H 2 + * (10) where * indicates the active site on the Na₂B₆ monolayer. It can be seen that the intermediate product of the HER process is only H*. As an efficient catalyst, its Gibb’s free energy should satisfy ΔG _H = 0 as much as possible. The most excellent Gibb’s free energy in HER of these X₂B₆ monolayer are obtained as Na₂B₆ monolayer, shown as Figure 6B, as about 0.64 eV, which is even lower than that of the MoSi₂N₄ (2.33 eV) [67]. Besides, in the OER reaction, the intermediate products are OH*, O* and OOH*. This process can be expressed as Equations 11–14 as follows: * + H 2 O   →   OH * + H + + e – (11) OH *   →   O * + H + + e – (12) O * + H 2 O   →   OOH * + H + + e – (13) OOH *   → * + O 2 + H + + e – (14)

One can see that the rate-determining step in the OER of the K₂B₆, Na₂B₆ and Rb₂B₆ monolayers is first step with the overpotentials about 1.78 eV, 2.19 eV and 2.28 eV, respectively, shown as Figure 6C. The insets in Figure 6C also demonstrated the adsorption configuration of intermediate. Moreover, the calculated OER catalytic activity of these X₂B₆ monolayers is also lower than that of the PtS₂/arsenene heterostructure (5.516 eV) and WSSe monolayer (2.39 eV). It is worth noting that the most stable HER and OER adsorption configuration of these system is demonstrated by binding energy (E _b), which is obtained as E _b = E _system–E _pure–E, where E _system, E _pure and E are the energy of the adsorbed X₂B₆, pure X₂B₆ monolayer and single intermediates, respectively. The lower binding energy imply the more stable configuration of the H*, OH*, O* and OOH*, showing as inset of Figures 6B, C.

In summary, the first-principle calculations are explore to predict the structural, electronic, mechanical, optical and catalytic properties systematically of the novel K₂B₆, Na₂B₆ and Rb₂B₆ monolayers. All these X₂B₆ monolayers present a stability structure, with an anisotropic Young’s modulus (296–406 N/m) and the Poisson’s ratio (0.36–0.35). Then, the ultra-narrow bandgap (0.42 eV) is obtained in the Na₂B₆ monolayer with high electron mobility as about 9942 cm². V⁻¹.s⁻¹. in decent transport direction. Furthermore, the excellent light absorption properties of the Na₂B₆ monolayer is also investigated. All these X₂B₆ monolayers suggest a low Gibb’s free energy in HER and OER, suggesting the potential applications as efficient nanodevice and catalyst.