⇦Chinese Physics Letters, 2023, Vol. 40, No. 1, Article code 017402 Theoretical Prediction of Superconductivity in Boron Kagome Monolayer: $M$B$_{3}$ ($M$ = Be, Ca, Sr) and the Hydrogenated CaB$_{3}$ Liu Yang (杨柳)1, Ya-Ping Li (李牙平)1, Hao-Dong Liu (刘浩东)1, Na Jiao (焦娜)1, Mei-Yan Ni (倪美燕)1, Hong-Yan Lu (路洪艳)1*, Ping Zhang (张平)1,2*, and C. S. Ting3 Affiliations 1School of Physics and Physical Engineering, Qufu Normal University, Qufu 273165, China 2Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 3Texas Center for Superconductivity and Department of Physics, University of Houston, Houston, Texas 77204, USA Received 23 November 2022; accepted manuscript online 26 December 2022; published online 3 January 2023 *Corresponding authors. Email: hylu@qfnu.edu.cn; zhang_ping@iapcm.ac.cn Citation Text: Yang L, Li Y P, Liu H D et al. 2023 Chin. Phys. Lett. 40 017402    Abstract Using first-principles calculations, we predict a new type of two-dimensional (2D) boride $M$B$_{3}$ ($M$ = Be, Ca, Sr), constituted by boron kagome monolayer and the metal atoms adsorbed above the center of the boron hexagons. The band structures show that the three $M$B$_{3}$ compounds are metallic, thus the possible phonon-mediated superconductivity is explored. Based on the Eliashberg equation, for BeB$_{3}$, CaB$_{3}$, and SrB$_{3}$, the calculated electron–phonon coupling constants $\lambda $ are 0.46, 1.09, and 1.33, and the corresponding superconducting transition temperatures $T_{\rm c}$ are 3.2, 22.4, and 20.9 K, respectively. To explore superconductivity with higher transition temperature, hydrogenation and charge doping are further considered. The hydrogenated CaB$_{3}$, i.e., HCaB$_{3}$, is stable, with the enhanced $\lambda $ of 1.39 and a higher $T_{\rm c}$ of 39.3 K. Moreover, with further hole doping at the concentration of $5.8\times 10^{11}$ hole/cm$^{2}$, the $T_{\rm c}$ of HCaB$_{3}$ can be further increased to 44.2 K, exceeding the McMillan limit. The predicted $M$B$_{3}$ and HCaB$_{3}$ provide new platforms for investigating 2D superconductivity in boron kagome lattice since superconductivity based on monolayer boron kagome lattice has not been studied before.
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 DOI:10.1088/0256-307X/40/1/017402 © 2023 Chinese Physics Society Article Text The kagome structure has been studied for more than 70 years, and its lattice consists of equivalent lattice points and bonds, forming corner-sharing triangles.[1] Because of the particularity of the lattice geometry and the intriguing electronic characteristics of flat band and Dirac cone in kagome lattice,[2-8] it shows magnetic,[9-12] topological,[13-16] or superconducting[17-19] properties. Particularly, superconductivity has been realized in three-dimensional (3D) materials containing kagome structures. For example, Li$_{2}$IrSi$_{3}$, with silicon atoms forming kagome lattices, is a superconductor with $T_{\rm c}$ of 3.8 K.[20,21] For RbBi$_{2}$, Bi atoms form kagome lattices, and its $T_{\rm c}$ is 4.15 K.[22] The $T_{\rm c}$ of LaRu$_{3}$Si$_{2}$ is 7 K, with Ru atoms forming kagome lattices.[23-25] Recently, a new family of kagome metals, $A$V$_{3}$Sb$_{5}$ ($A$ = K, Ru, Cs), has been synthesized in experiment, with V atoms forming kagome lattices.[26] They exhibit rich physical properties, such as topology, charge density wave (CDW) phase, and superconductivity.[27-30] The superconducting $T_{\rm c}$'s of $A$V$_{3}$Sb$_{5}$ ($A$ = K, Ru, Cs) are 0.93,[31] 0.92,[32] and 2.5 K,[33] respectively. Superconductivity in 2D materials with kagome lattice has also attracted more and more attention. Once realized, it may be not only of scientific significance, but also of potential applications in constructing nano-superconducting devices. It was predicted that 2D phosphorous with buckling P atoms forming kagome lattice is a semiconductor, but can be a superconductor with $T_{\rm c}$ of 9 K under appropriate hole doping.[34] Boron atom has three valence electrons. Thus, borophene shows various configurations with boron atoms forming triangular lattice structure or with hexagonal voids,[35-38] which may also be a good choice for a kagome material. However, the pure boron kagome lattice in a monolayer is dynamically unstable.[39] Recently, a new structure containing bilayer boron kagome lattice with a layer of metal atoms sandwiched between them, i.e., $M$B$_{6}$ ($M$ = Mg, Ca, Sc, Ti), was predicted to be stable, and they can be superconductors with the highest $T_{\rm c}$ of 22.6 K and the corresponding $\lambda$ of 0.87 in CaB$_{6}$.[40,41] Meanwhile, h-MnB$_{3}$, containing two boron kagome layers with two layers of Mn atoms sandwiched between them, was also predicted to be a superconductor with $T_{\rm c}$ of 24.9 K, and the $T_{\rm c}$ can be increased to 34 K under 2% tensile strain.[39] In the above-mentioned bilayer boron kagome structure, the electron transferred from the sandwiched metal atoms to boron layers stabilizes the kagome lattice. Till now, the stability and superconductivity based on monolayer boron kagome lattice have not been studied. Whether certain atoms could make boron kagome monolayer stable and exhibit superconductivity is worth exploring. It is known that the adsorption of metal atoms on graphene can modify the electronic structure, enhance electron–phonon coupling (EPC), and induce superconductivity. For example, Li and Ca deposited graphene, i.e., LiC$_{6}$ and CaC$_{6}$, are superconductors with $T_{\rm c}$ of 8.1 and 1.4 K, respectively.[42] The $T_{\rm c}$ of Al deposited graphene AlC$_{8}$ is 22.2 K under proper hole doping and stretching.[43] Thus, it is meaningful to study the stability and superconductivity of boron kagome monolayer with metal atoms adsorbed on it. In this work, based on first-principles calculations, we predict that Be, Ca, and Sr atoms adsorbed on the monolayer boron kagome lattice can stabilize the lattice and turn the materials to be superconductors. For the obtained $M$B$_{3}$ ($M$ = Be, Ca, Sr), the EPC constants $\lambda$ are 0.46, 1.09, and 1.33, and the corresponding $T_{\rm c}$'s are 3.2, 22.4, and 20.9 K, respectively. Hydrogenation is also considered to be an effective way to modulate the electronic structure and superconductivity of 2D materials. A series of studies on 2D hydride high-temperature superconductors at atmospheric pressure have been carried out in recent years. For example, the hole-doped graphane (fully hydrogenated graphene) can be a superconductor with $T_{\rm c}$ of 90 K.[44] The $T_{\rm c}$ of hydrogenated MgB$_{2}$ is predicted to be 67 K, which can be increased to more than 100 K by applying 5$\%$ biaxial tensile strain.[45] Our recent study shows that hydrogenation can turn the 2D phosphorus carbide PC$_{3}$ from a semiconductor to a superconductor with $T_{\rm c}$ of 31.0 K, which can be boosted to 57.3 K by applying 3$\%$ biaxial tensile strain.[46] The hydrogenated 2H-Mo$_{2}$C$_{3}$, i.e., 2H-Mo$_{2}$C$_{3}$H$_{2}$, is a 2D superconductor with $T_{\rm c}$ of 53 K.[47] Therefore, it is of great significance to study the effects of hydrogenation on the electronic properties and possible high temperature superconductivity of $M$B$_{3}$ ($M$ = Be, Ca, Sr). The calculated results show that, after hydrogenation, HBeB$_{3}$ is a semiconductor, and HSrB$_{3}$ is thermodynamically unstable at room temperature, whereas HCaB$_{3}$ retains its metallic property and is still thermodynamic stable at 400 K. Thus, we further study the superconductivity of HCaB$_{3}$. The results show that HCaB$_{3}$ is also dynamically stable, the EPC strength and the $T_{\rm c}$ are increased to 1.39 and 39.3 K, respectively. Moreover, by doping 0.06 hole per unit cell, the $T_{\rm c}$ of HCaB$_{3}$ can be further increased to 44.2 K, exceeding the McMillan limit. Since superconductivity based on a single boron kagome lattice has not been studied before, the predicted $M$B$_{3}$ and HCaB$_{3}$ provide new platforms for investigating 2D superconductivity in boron kagome lattice. Firstly, we show the structures of $M$B$_{3}$ ($M$ = Be, Ca, Sr) and hydrogenated CaB$_{3}$ (HCaB$_{3}$), and prove their stabilities. Figure 1(a) shows the structure of CaB$_{3}$, where boron atoms form kagome lattice and Ca atoms lie above the center of the boron hexagons. The structures of BeB$_{3}$ and SrB$_{3}$ are shown in Figs. S1(a) and S1(b) in the Supplemental Material. The space group of these structures is $P6mm$. After full optimization, the detailed structure parameters are listed in Table 1. The lattice constant of CaB$_{3}$ is 3.433 Å, the B–B bond length is 1.716 Å, and the height of Ca to the boron kagome layer is 2.14 Å. The electron localization function (ELF) proves that this structure contains strong B–B covalent bonds as shown in Fig. S3(5), and the ELF value between B and Ca atoms is very small as shown in Figs. S3(7–8), however, the Bader charge analysis shows 0.728$e$ transfer from each Ca atom to the boron kagome layer, indicating that B and Ca atoms form ionic bonds. Moreover, charge density difference is also calculated as shown in Fig. 1(e), which clearly shows charge transfer from Ca to B. The above three aspects together prove that the Ca atoms lose electrons and form B–Ca bonds with the boron plane. For BeB$_{3}$ and SrB$_{3}$, similar results can be obtained. It is known that the pure boron kagome layer is dynamically unstable due to the electron deficiency of boron.[40] The above results indicate that the captured electrons from metal atoms and the formation of the $M$–B bonds can stabilize the boron kagome layer when the metal atoms are adsorbed on the boron kagome layer.
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Fig. 1. Top and side views of the structures of (a) CaB$_{3}$ and (b) HCaB$_{3}$. Green (blue, pink) spheres represent B (Ca, H) atoms and the unit cell is shown. Here $h$ is the perpendicular distance between Ca and B in the kagome lattice. The variation of the free energy in the AIMD simulations during the time scale of 6 ps along with the last frame of photographs at 400 K for CaB$_{3}$ (c) and HCaB$_{3}$ (d). Charge density difference for (e) CaB$_{3}$ and (f) HCaB$_{3}$. The color blue (yellow) represents the area where electrons are lost (obtained).
Table 1. Optimized lattice constant $a$ (Å), metal-boron-kagome-plane distance $h$ (Å), B–B bond length $l$ (Å), value of the transferred charge from the metal atoms $n$ ($e$) by Bader charge analysis, cohesive energy $E_{\rm coh}$ (eV/atom), logarithmically averaged phonon frequency $\omega_{\log}$ (K), total electron–phonon coupling (EPC) constant $\lambda$, and superconducting transition temperature $T_{\rm c}$ (K) for $M$B$_{3}$ ($M$ = Be, Ca, Sr) and HCaB$_{3}$.
$a$ $h$ $l$ $n $ $E_{\rm coh}$ $\omega_{\log}$ $\lambda$ $T_{\rm c}$
BeB$_{3}$ 3.464 0.65 1.732 1.544 $-5.55$ 355.3 0.47 3.2
CaB$_{3}$ 3.433 2.14 1.716 0.728 $-5.05$ 284.0 1.09 22.4
SrB$_{3}$ 3.465 2.30 1.732 0.624 $-4.95$ 207.2 1.33 20.9
HCaB$_{3}$ 3.436 2.17 1.715 1.331 $-4.89$ 371.5 1.39 39.3
After hydrogenation, the space group of HCaB$_{3}$ is changed to $P3m1$. As shown in Fig. 1(b), H atoms locate above the hollow position of the B triangle. From the side view, it is clearly seen that the boron kagome layer is still flat with Ca atoms above it and H atoms above Ca atoms. The lattice constant of HCaB$_{3}$ is 3.436 Å, the distance between Ca and boron kagome plane is 2.17 Å, and the B–B bond length is 1.715 Å. The distance between H and Ca is 0.966 Å. Compared with CaB$_{3}$, the lattice constant of HCaB$_{3}$ increases slightly, and the perpendicular distance between Ca and B also increases slightly, as shown in Table 1. Bader charge analysis shows that each Ca atom loses 1.331$e$ charges, including 0.628$e$ to boron kagome layer and 0.703$e$ to H atoms. The ELF values in Figs. S3(15–16) show that the Ca atoms lose electrons and form Ca–B and Ca–H ionic bonds. The stabilities of $M$B$_{3}$ and HCaB$_{3}$ are derived from three aspects, i.e., cohesive energy, ab initio molecular dynamics (AIMD), and phonon spectra. The cohesive energies are calculated using the following formulas: $E_{\rm coh}=[{E_{\rm MB_{3}}-(E_{\rm M}+3E_{\rm B})]}/4$ for $M$B$_{3}$, and $E_{\rm coh}=[{E_{\rm HCaB_{3}}-(E_{\rm Ca}+3E_{\rm B}+E_{\rm H})]}/5$ for HCaB$_{3}$, where $E_{\rm MB_{3}}$, $E_{\rm HCaB_{3}}$, $E_{\rm M}$, $E_{\rm B}$, and $E_{\rm H}$ are the total energies of $M$B$_{3}$, HCaB$_{3}$, isolated metal, boron, and hydrogen atoms, respectively. According to this definition, a more negative $E_{\rm coh}$ value indicates higher thermodynamic stability. As listed in Table 1, the calculated cohesive energies of BeB$_{3}$, CaB$_{3}$, SrB$_{3}$, and HCaB$_{3}$, are $-5.55$, $-5.05$, $-4.95$, and $-4.89$ eV/atom, respectively, which are lower than some theoretically predicted and later experimentally fabricated 2D materials, such as Cu$_{2}$Si ($-3.46$ eV/atom),[48,49] Ni$_{2}$Si ($-4.80$ eV/atom),[50] and Cu$_{2}$Ge ($-3.17$ eV/atom).[51] These results of cohesive energies indicate that the predicted $M$B$_{3}$ and HCaB$_{3}$ could be synthesized under appropriate experimental conditions. For the AIMD simulations, we adopt a $5\times5\times1$ supercell to minimize the effect of the periodic boundary condition. The variation of free energy in the AIMD simulations within 6 ps and the last frame of the photographs are exhibited in Fig. 1(c) for CaB$_{3}$ and Fig. 1(d) for HCaB$_{3}$, respectively. CaB$_{3}$ and HCaB$_{3}$ maintain the structural integrity even at 400 K and the energy fluctuates around $-503$ and $-616$ eV, proving their thermodynamical stability. Moreover, the phonon spectra of CaB$_{3}$ and HCaB$_{3}$ exhibit no imaginary frequency, as shown in Figs. 3(a) and 3(e), indicating that they are dynamically stable. In the main text, we only discuss CaB$_{3}$ and HCaB$_{3}$, and the results of BeB$_{3}$ and SrB$_{3}$ are discussed in the Supplemental Material. The structures, AIMD simulations and phonon spectra of BeB$_{3}$ and SrB$_{3}$ are shown in Figs. S1 and S5 in the Supplemental Material, proving that they are also stable.
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Fig. 2. Orbital-projected band structures of (a) B and (b) Ca for CaB$_{3}$ along the high-symmetry line $\varGamma$–$M$–$K$–$\varGamma$. (c) Total DOS of CaB$_{3}$, B and Ca. (d) Orbital-projected DOS of CaB$_{3}$. Here (e)–(h) are similar with (a)–(d) but for HCaB$_{3}$.
Figure 2 shows the electronic structures of CaB$_{3}$ and HCaB$_{3}$, including the orbital-resolved band structure, electronic DOS and orbital-projected DOS. For CaB$_{3}$, Figs. 2(a) and 2(b) show the orbital-resolved band structure along high-symmetry line $\varGamma$–$M$–$K$–$\varGamma$ for B and Ca, respectively, showing that CaB$_{3}$ is a metal. It is known that for the Kagome model, when only considering the single orbital model and the nearest hopping between the kagome sites, a totally flat band can appear.[52,53] However, according to the first-principles calculations, real material-systems contain multi-orbitals, and terms beyond the nearest hopping, these would make the “flat band” dispersive and not so flat. There exist publications that demonstrate this feature, see, e.g., Refs. [54-57]. For CaB$_{3}$, the relative flat band can be found at about 1.5 eV from $M$ to $K$ in Figs. 2(a) and 2(b). In Fig. 2(c), the total DOS of each element shows that the contribution by B and Ca atoms are comparable at the Fermi level. According to Fig. 2(d), the most contribution is from B-$p_{z}$ orbitals, followed by the degenerated Ca-$d_{xy}$ and Ca-$d_{x^{2}-y^{2}}$ orbitals. Furthermore, Ca-$d_{xz}$, Ca-$d_{yz}$, B-$p_{x}$, and B-$p_{y}$ orbitals also show a large contribution around the Fermi level. These results indicate that there is a strong hybridization between Ca and B atoms at the Fermi energy $E_{\rm F}$. The valence electrons of Ca atom provided by the PBE pseudopotential are in $3s$, $3p$, and $4s$ orbitals. Bader charge analysis shows that Ca atoms lose electrons in CaB$_{3}$. Some of the electrons are transferred from $4s$ orbital of Ca to the B layer, which stabilize the kagome lattice, and the others are transferred to the $3d$ orbital of Ca, which contribute to the $3d$ electron states at the Fermi level.
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Fig. 3. Phonon dispersion weighted by the vibration modes of the atoms for (a) CaB$_{3}$ and (e) HCaB$_{3}$, and phonon dispersion weighted by the magnitude of EPC $\lambda_{\boldsymbol{q}\nu}$ for (b) CaB$_{3}$ and (f) HCaB$_{3}$. The insets show the vibration modes for the prominent $\lambda_{\boldsymbol{q}\nu}$ I, II, and III. Atom-projected phonon DOS for (c) CaB$_{3}$ and (g) HCaB$_{3}$. Eliashberg spectral function $\alpha^{2}F(\omega)$ and cumulative frequency dependence of EPC $\lambda(\omega)$ for (d) CaB$_{3}$ and (h) HCaB$_{3}$.
For HCaB$_{3}$, it is also a metal, with two bands crossing the Fermi level. One band crosses the Fermi level along $\varGamma$–$M$ and $M$–$K$, respectively, and the other one crosses the Fermi level along $K$–$\varGamma$, as shown in Figs. 2(e) and 2(f). Flat bands can be found at about 2.5 eV from $M$ to $K$ as well as $-3.5$ eV along the high-symmetry line $\varGamma$–$M$–$K$–$\varGamma$. From the DOS of HCaB$_{3}$ in Figs. 2(g) and 2(h), it is shown that the states at the Fermi level are mainly contributed by B and Ca atoms, and the contribution of H atoms is small. The PDOS of HCaB$_{3}$ is shown in Fig. 2(h), which proves that the $2p$ orbital of B atoms contributes mostly to the bands near the Fermi level, and then is the $3d$ orbital of Ca atoms, the last is the $1s$ orbital of H atoms. Below the Fermi level, there is a Van Hove singularity peak at $-0.1$ eV, which mainly comes from the flat band below the Fermi level near the high-symmetry point $M$ along $M$–$K$ direction. Compared with CaB$_{3}$, the band structure of SrB$_{3}$ is very similar, as presented in Fig. S4 of the Supplemental Material, because of the similar valence electrons distribution. However, the band structure of BeB$_{3}$ is different. The detailed electronic structures of BeB$_{3}$ and SrB$_{3}$ are shown in the Supplemental Material. To explore the possible phonon-mediated superconductivity, we study the phonon dispersion, phonon DOS, Eliashberg spectral function $\alpha^{2}F(\omega)$, and cumulative frequency dependence of EPC $\lambda(\omega)$ of CaB$_{3}$ and HCaB$_{3}$, which are shown in Fig. 3. For CaB$_{3}$, there are four atoms in the unit cell, leading to twelve phonon modes. It includes three acoustic phonon modes and nine optical phonon modes, which are shown in Figs. 3(a) and 3(b). The phonon spectrum shows a wide range of frequency extending up to about 1135 cm$^{-1}$. From the phonon DOS in Fig. 3(c), the vibration modes can be divided into two parts, the low-frequency range ($ < $ 335 cm$^{-1}$) contributed by both the vibrations of Ca and B, and the high-frequency range ($> $ 335 cm$^{-1}$) dominated by vibrations of B atoms. The cumulative frequency dependence of EPC $\lambda(\omega)$ in Fig. 3(d) shows that the EPC mainly comes from the low-frequency vibration modes of CaB$_{3}$. Combined with Fig. 3(a), the following results can be obtained. In the rang from 0 to 196 cm$^{-1}$, the vibrations are mainly acoustic phonon modes. The coupling between electrons and these phonons (mainly from the in-plane vibrations of Ca atoms) contributes 55${\%}$ of the total EPC ($\lambda=1.09$). From 196 to 438 cm$^{-1}$, the coupling between electrons and the out-of-plane phonons of B contributes 36${\%}$ of the total EPC ($\lambda=1.09$). In the high-frequency range above 438 cm$^{-1}$, the $\lambda(\omega)$ increases slowly, indicating that the coupling between electrons and in-plane phonons of B is very weak. Moreover, from the EPC magnitude $\lambda_{\boldsymbol{q}\nu}$ in Fig. 3(b), the strongest coupling appears along the $K$–$\varGamma$ at the frequency of 119 cm$^{-1}$ (mode I), which originates from the coupling between electrons and in-plane phonons of Ca atoms and a small portion of in-plane and out-of-plane phonons of B atoms, as shown in the inset of Fig. 3(b). Then, we study the effect of hydrogenation on superconductivity of CaB$_{3}$. For HCaB$_{3}$, there are five atoms in the unit cell, leading to fifteen phonon modes, which include three acoustic modes and twelve optical modes shown in Figs. 3(e) and 3(f). The phonon spectra show a wide range of frequency extending up to about 1096 cm$^{-1}$. Compared with the phonon spectrum of CaB$_{3}$ in Fig. 3(a), the most obvious change is the appearance of certain branches originating from hydrogen in high frequency, because of the small mass of H. The Eliashberg spectral function and $\lambda(\omega)$ of HCaB$_{3}$ in Fig. 3(h) indicate that its EPC also mainly origins from the low-frequency part. Although H atoms show small phonon DOS in the low-frequency part, it plays an important role in the EPC. For the energy below 481 cm$^{-1}$, the EPC mainly originates from the coupling between electrons and the in-plane phonons of H as well as out-of-plane phonons of B atoms, contributing 85${\%}$ of the total EPC ($\lambda=1.39$). For the frequencies above 730 cm$^{-1}$, the phonon DOS of H atoms is significant, as shown in Fig. 3(g). In this range, the coupling between electrons and in-plane phonons of B as well as in-plane and out-of-plane phonons of H atoms contributes 11${\%}$ of the total EPC ($\lambda=1.39$). Compared with $\lambda(\omega)$ of CaB$_{3}$, $\lambda(\omega)$ of HCaB$_{3}$ shows a significant increase in the frequency range from 730 to 1000 cm$^{-1}$. In Fig. 3(f), there are two large $\lambda_{\boldsymbol{q}\nu}$, contributing greatly to the EPC. One is along $\varGamma$–$M$ at the frequency 262 cm$^{-1}$ (mode II), and the other is around $\varGamma$ point at the frequency 343 cm$^{-1}$ (mode III). The insets show the vibration for the two modes, both mainly from the out-of-plane vibration of B atoms.
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Fig. 4. (a) Phonon spectra for the pristine (in black) and hole-doped HCaB$_{3}$ with $p = 0.03$ hole/cell (in blue) and $p = 0.06$ hole/cell (in red). (b) The corresponding phonon DOS for the three cases in (a).
It is known that electron or hole doping can modulate the band structure and also the superconducting properties. For example, graphane is a semiconductor, but can be a superconductor after hole doping.[44] For the total DOS of HCaB$_{3}$ in Fig. 2(g), it can be seen that the DOS at the energy slightly below the Fermi level is larger than that at the Fermi energy. Therefore, a higher $T_{\rm c}$ is anticipated if the Fermi level moves down. Here, we study the properties of HCaB$_{3}$ doped with $p = 0.03$ and $p = 0.06$ hole/cell. We have fully relaxed the lattice at the two doping concentrations. The hole doping concentrations are $2.919 \times 10^{11}$ and $5.841 \times 10^{11}$ cm$^{-2}$ for 0.03 and 0.06 hole/cell, respectively. In experiment, doping level up to more than $10^{14}$ cm$^{-2}$ can be achieved by electrolytic gating.[58] Thus, the doping levels under concern are experimentally accessible. The phonon spectra for the pristine and doped cases are shown in Fig. 4. There is also no imaginary frequency for the doped cases, indicating that the structures are stable. Compared with the pristine HCaB$_{3}$, there is almost no change in the phonon spectrum of $p = 0.03$ hole/cell case. For the $p = 0.06$ hole/cell case, the low-frequency part of the phonon spectrum is not changed, but at high frequency, the frequency of modes originating from the in-plane vibration of H increases and that from the in-plane vibration of B decreases, compared with the vibration modes analysis of the pristine HCaB$_{3}$ shown in Fig. 3(e). Figure 5(a) shows the electronic DOS of the pristine and hole-doped HCaB$_{3}$ with $p = 0.03$ and $p = 0.06$ hole/cell. After hole doping, the DOS at the Fermi level increases. Figures 5(b), 5(c), and 5(d) are the Eliashberg spectral function $\alpha^{2}F(\omega)$ and cumulative frequency dependence of EPC $\lambda(\omega)$ for the three cases. After hole doping, it shows that EPC of HCaB$_{3}$ is still mainly contributed by the coupling between electrons and the low-frequency phonon. After $p = 0.03$ and $p = 0.06$ hole/cell doping, ${\lambda}$ increases from 1.39 for the pristine case to 1.45 and 1.58, and $T_{\rm c}$ increases from 39.3 K for the pristine case to 41.6 and 44.2 K, respectively, exceeding the McMillan limit. Therefore, hole doping is an effective tool to the increase of $T_{\rm c}$. For BeB$_{3}$ and SrB$_{3}$, their $\lambda$'s are 0.47 and 1.33, and the corresponding $T_{\rm c}$'s are 3.2 and 20.9 K, respectively, which are shown in detail in the Supplemental Material.
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Fig. 5. (a) Total electronic DOS for the pristine and hole-doped HCaB$_{3}$ with $p = 0.03$ and $p = 0.06$ hole/cell. [(b), (c), (d)] The Eliashberg spectral function $\alpha^{2}F(\omega)$ and cumulative frequency dependence of EPC $\lambda(\omega)$ for the three cases in (a).
Before conclusion, we make some discussions. Although both CaB$_{3}$ and HCaB$_{3}$ are possible phonon-mediated superconductors, the origin of EPC shows some differences. For CaB$_{3}$, the EPC mainly originates from the coupling between electrons of B and Ca atoms and in-plane phonons of Ca atoms. After hydrogenation, H atoms contribute to both the electronic DOS at the Fermi energy and the phonon vibration modes, especially in the high-frequency range. Compared with CaB$_{3}$, the EPC of HCaB$_{3}$ increases from 1.09 to 1.39 and the $T_{\rm c}$ increases from 22.4 K to 39.3 K. By further hole doping, the electronic DOS at the Fermi energy increases, and the $T_{\rm c}$ is further increased to 44.2 K at the doping level $p = 0.06$ hole/cell. Therefore, our work enriches the superconductivity database of 2D monolayer boron kagome materials, and also proves that hydrogenation and doping are effective methods to regulate the superconductivity of 2D materials, which is worthy for us to make more efforts in this field. In conclusion, based on first-principles calculations, we have predicted a new type of 2D boride with boron atoms forming a kagome layer: monolayer $M$B$_{3}$ ($M$ = Be, Ca, Sr) and the hydrogenated CaB$_{3}$. Their dynamical and thermodynamical stabilities are proved by phonon spectrum and AIMD, respectively. Our results show that the calculated EPC $\lambda$'s for BeB$_{3}$/CaB$_{3}$/SrB$_{3}$ are 0.47/1.09/1.33, and the corresponding $T_{\rm c}$'s are 3.2/22.4/20.9 K. Furthermore, we have investigated the effect of hydrogenation and charge doping on the superconductivity of CaB$_{3}$. For HCaB$_{3}$, the $\lambda$ is 1.39 and the $T_{\rm c}$ is increased to 39.3 K. After hole doping at the concentration of $5.841 \times 10^{11}$ cm$^{-2}$, the $T_{\rm c}$ of HCaB$_{3}$ can be further increased to 44.2 K, exceeding the McMillan limit. The above results indicate that hydrogenation and doping can effectively modulate the electronic and phonon properties, so as to increase the $T_{\rm c}$. Since superconductivity based on a single boron kagome lattice has not been studied, the predicted $M$B$_{3}$ and HCaB$_{3}$ provide new platforms for investigating 2D superconductivity in boron kagome lattice. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12074213, 11574108, and 12104253), the Major Basic Program of Natural Science Foundation of Shandong Province (Grant No. ZR2021ZD01), the Project of Introduction and Cultivation for Young Innovative Talents in Colleges and Universities of Shandong Province, and the Texas Center for Superconductivity at University of Houston, the Robert A. Welch Foundation (Grant No. E-1146). 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