High-temperature superconductivity in ${\mathrm{SrB}}_3{\mathrm{C}}_3$ and ${\mathrm{BaB}}_3{\mathrm{C}}_3$ predicted from first-principles anisotropic Migdal-Eliashberg theory

Very recently, carbon-boron clathrate $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ has been successfully synthesized, in which carbon and boron atoms form $s{p}^3$-bonded truncated octahedral cages. Interestingly, the $s{p}^3$-hybridized $\sigma$-bonding bands are partially occupied. This may drive $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ into a superconducting state, like boron-doped diamond. By means of density functional first-principles calculations and Wannier interpolation technique, we have investigated the electron-phonon coupling and phonon-mediated superconductivity in $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$. Our calculations reveal that there exists strong coupling between $s{p}^3$-hybridized $\sigma$-bonding bands and boron-associated $E_g$ phonon modes. Based on the Migdal-Eliashberg theory, we self-consistently solve the anisotropic Eliashberg equations. It is found that $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ is a single-gap superconductor, with superconducting transition temperature being 40 K. The anisotropic ratio of superconducting energy gap is computed to be 32.8%. Further replacing Sr with Ba, the transition temperature can be boosted to 43 K in $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$ due to phonon softening. These findings suggest that $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$ are phonon-mediated high-temperature anisotropic $s$-wave superconductors.

Figure 1

Crystal structures for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. (a) Three-dimensional view. (b) View along the [100] direction. The cyan, green, and red balls represent strontium/barium, boron, and carbon atoms, respectively. The solid black line denotes the unit cell.

Figure 2

(a)–(k) Electronic structure for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$. (a) Band structure. The red lines and blue circles denote the bands obtained by first-principles calculation and interpolating the real-space Hamiltonian, respectively. The Fermi level is set to zero. (b) Total and orbital-resolved DOS. The unit for DOS is states/spin/eV/cell. (c)–(e) Fermi surfaces, whereas the colors stand for the strength of EPC ${\lambda }_{\mathbf{k}n}$. (f)–(h) The weights of C-$2p$ orbitals on the Fermi surfaces. (i)–(k) The weights of B-$2p$ orbitals on the Fermi surfaces. (l)–(v) Electronic structure for $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$.

Figure 3

Lattice dynamics of $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. Phonon spectrum with a color representation of ${\lambda }_{\mathbf{q}v}{\omega }_{\mathbf{q}v}$ for (a) $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and (c) $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. $E_g$ are the strongly coupled phonon modes. Projected phonon DOS generated by quasiharmonic approximation for (b) $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and (d) $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. (e)–(f) Phonon displacements for double degenerate $E_g$ modes at the $\mathrm{\Gamma }$ point. The displacements are denoted by blue arrows.

Figure 4

(a) Phonon DOS $F\left(\omega \right)$, Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$, and accumulated $\lambda \left(\omega \right)$ for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$. (b) Normalized superconducting gap distributions of $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ at different temperatures. (c) $F\left(\omega \right), {\alpha }^{2}F\left(\omega \right)$, and $\lambda \left(\omega \right)$ for $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. (d) Normalized superconducting gap distributions of $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. $\lambda \left(\omega \right)$ is computed using the formula $2{\int }_0^{\omega }\frac{1}{{\omega }^{\prime }}{\alpha }^{2}F\left({\omega }^{\prime }\right)d{\omega }^{\prime }$. The graduation of $F\left(\omega \right)$ is omitted for clarity. The violet lines in (c) and (d) represent the average ${\mathrm{\Delta }}_{\mathbf{k}n}$ for each temperature. (e)–(g) Distributions of superconducting gap ${\mathrm{\Delta }}_{\mathbf{k}n}$ on Fermi surfaces for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ at 5 K. (h)–(j) Distributions of superconducting gap ${\mathrm{\Delta }}_{\mathbf{k}n}$ on Fermi surfaces for $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$ at 5 K. The screened Coulomb potential ${\mu }^{*}$ is set to 0.1.

Figure 5

Comparisons of nesting function $\xi \left(\mathbf{q}\right)$, EPC matrix element weighted nesting function $\gamma \left(\mathbf{q}\right)$, and q-resolved EPC constant $\lambda \left(\mathbf{q}\right)$ for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$. In these two compounds, $\xi \left(\mathbf{q}\right)$ and $\gamma \left(\mathbf{q}\right)$ are respectively normalized by $\xi \left(\mathrm{\Gamma }\right)$ and $\gamma \left(\mathrm{\Gamma }\right)$ of $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$.

Figure 6

Convergence test of EPC constant $\lambda$ for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ (a) and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$ (b). Spatial decay of electronic Hamiltonian in the Wannier representation for $\mathrm{Sr}{\mathrm{B}}_3{\mathrm{C}}_3$ (c) and $\mathrm{Ba}{\mathrm{B}}_3{\mathrm{C}}_3$ (d). Here, the absolute value of hopping integral of the Hamiltonian is normalized by the maximal one.