Prediction of phonon-mediated superconductivity in borophene

Superconductivity in two-dimensional compounds is widely studied, not only because of its application in constructing nano-superconducting devices, but also for general scientific interest. Very recently, borophene (a two-dimensional boron sheet) has been successfully grown on the Ag(111) surface, through direct evaporation of a pure boron source. The experiment unveiled two types of borophene structures, namely ${\beta }_{12}$ and ${\chi }_3$. Herein, we employed density-functional first-principles calculations to investigate the electron-phonon coupling and superconductivity in both structures of borophene. The band structures of ${\beta }_{12}$ and ${\chi }_3$ borophenes exhibit inherent metallicity. We found that electron-phonon coupling constants in the two compounds are larger than that in ${\mathrm{MgB}}_2$. The superconducting transition temperatures were determined to be 18.7 K and 24.7 K through the McMillian-Allen-Dynes formula. These temperatures are much higher than the theoretically predicted 8.1 K and experimentally observed 7.4 K superconductivity in graphene. Our findings will enrich nano-superconducting device applications and boron-related materials science.

Figure 1

Top views of borophenes. (a) ${\beta }_{12}$ borophene. (b) ${\chi }_3$ borophene. The green balls and red hollow circles represent the boron atoms and vacancy sites, respectively. The rectangle and rhombus enclosed by solid black lines denote the unit cells for ${\beta }_{12}$ and ${\chi }_3$ borophenes.

Figure 2

Calculated electronic structures of ${\beta }_{12}$ and ${\chi }_3$ borophenes. (a) The band structure of ${\beta }_{12}$ (left panel) and the Brillouin zone and Fermi surfaces (right panel). (b) The band structure of ${\chi }_3$ (left panel) and the Brillouin zone and Fermi surfaces (right panel). The blue lines represent band structures given by first-principles calculation. The red curves are obtained by interpolation of maximally localized Wannier functions (MLWFs). The Fermi energy is set to zero. The high-symmetry points used in the band-structure calculations are labeled in the corresponding Brillouin zone.

Figure 3

Lattice dynamics of ${\beta }_{12}$ borophene. (a) Phonon band structures, in which the linewidths for phonon mode $\mathbf{q}\nu$ (i.e., ${\gamma }_{\mathbf{q}\nu }$) are represented by the widths of red lines. (b) Phonon density of states (DOS) $F\left(\omega \right)$ and Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. (c) and (d) Vibrational pattern for two $A_g$ phonon modes at $\mathrm{\Gamma }$. The red arrows and their lengths denote the directions and relative amplitudes of these vibrational modes, respectively.

Figure 4

Lattice dynamics of ${\chi }_3$ borophene. (a) Phonon band structures, in which the linewidths for phonon mode $\mathbf{q}\nu$ (i.e., ${\gamma }_{\mathbf{q}\nu }$) are represented by the widths of red lines. (b) Phonon DOS $F\left(\omega \right)$ and Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$. (c)–(f) Vibrational patterns for important phonon modes. The red arrows and their lengths denote the directions and relative amplitudes of these vibrational modes, respectively.

Figure 5

Positive DCD (isovalue: $2.8\times {10}^{-2}e/{\mathrm{bohr}}^3$) (a), negative DCD (isovalue: $-8.0\times {10}^{-3}e/{\mathrm{bohr}}^3$) (b), orbital-resolved and phonon-perturbed band structures (c), and Fermi surfaces perturbed by $B_{3g}$ phonon mode (d) in ${\beta }_{12}$ borophene. Positive DCD (isovalue: $2.45\times {10}^{-2}e/{\mathrm{bohr}}^3$) (e), negative DCD (isovalue: $-8.0\times {10}^{-3}e/{\mathrm{bohr}}^3$) (f), orbital-resolved and phonon-perturbed band structures (g), and Fermi surfaces perturbed by $B_{3g}$ phonon mode (h) in ${\chi }_3$ borophene. The widths of blue and green lines in the band structures stand for the weights of $2s+2p_x+2p_y$ and $2p_z$ orbitals of boron in given Kohn-Sham state. The red lines represent the band structures perturbed by $B_{3g}$ phonon modes at the $\mathrm{\Gamma }$ point. The obvious deviations in band structures before and after phonon perturbation are enclosed by gray shadows.

Figure 6

Spatial decay of electronic Hamiltonian and dynamical matrix in the Wannier representation. (a) and (c) represented decay of electronic Hamiltonian for ${\beta }_{12}$ and ${\chi }_3$ borophene. Decay of dynamical matrix for ${\beta }_{12}$ and ${\chi }_3$ borophene are shown in (b) and (d), respectively.

Figure 7

Convergence test of EPC constant versus smearing parameter $\sigma$ in ${\beta }_{12}$ borophene. In the calculation, fine phonon grid was chosen to be $240\times 160\times 1$. The inset shows the convergence of $\lambda$ for $\sigma =0.015$ eV, with increasing density of k mesh.

Figure 8

Convergence test of EPC constant versus smearing parameter $\sigma$ in ${\chi }_3$ borophene. In the calculation, fine phonon grid was chosen to be $180\times 180\times 1$.

Figure 9

(a) Ground-state structure of ${\chi }_3/\text{SL-Ag}$(111). The gray balls represent Ag atoms. The solid lines denote the unit cell. (b) Band structures of ${\chi }_3/\text{SL-Ag}$(111) obtained by first-principles calculation (blue lines) and interpolation of MLWFs (red lines).

Figure 10

Phonon DOS and Eliashberg spectral function for ${\chi }_3/\text{SL-Ag}$(111).