Van Hove singularity induced phonon-mediated superconductivity above 77 K in hole-doped ${\mathrm{SrB}}_3{\mathrm{C}}_3$

To design a phonon-mediated superconductor with transition temperature above the liquid-nitrogen temperature under ambient pressure, we align the Fermi level of compound ${\mathrm{SrB}}_3{\mathrm{C}}_3$ to a Van Hove singularity by introducing 0.4 hole/f.u., namely ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. Based on density functional first-principles calculations and the Wannier interpolation technique, the electronic structure, lattice dynamics, and electron-phonon coupling (EPC) of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ are investigated. Both the enlarged density of states at the Fermi level and the softened phonons play important roles to enhance the EPC. By solving the anisotropic Eliashberg equations, we find that the transition temperature of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ is 83 K, successfully exceeding the liquid-nitrogen temperature at ambient pressure condition. Moreover, our calculations reveal that there is a single-gap to two-gap superconductivity transition in comparison with ${\mathrm{SrB}}_3{\mathrm{C}}_3$.

Figure 1

Electronic structure for ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. (a) Band structure. The Fermi level is set to zero. (b) Total and orbital-resolved density of states. Distributions of B-$2p$ orbitals (c)–(f) and C-$2p$ orbitals (g)–(j) on the Fermi surfaces.

Figure 2

Lattice dynamics of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. (a) Phonon spectrum, with a color representation of ${\lambda }_{\mathbf{q}v}{\omega }_{\mathbf{q}v}$. (b) Phonon DOS $F\left(\omega \right)$. (c) Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ and accumulated $\lambda \left(\omega \right)$. The accumulated $\lambda \left(\omega \right)$ is computed using the formula $2{\int }_0^{\omega }\frac{1}{{\omega }^{\prime }}{\alpha }^{2}F\left({\omega }^{\prime }\right)d{\omega }^{\prime }$. (d) Comparison of Fermi surfaces nesting between ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ and ${\mathrm{SrB}}_3{\mathrm{C}}_3$. $\xi \left(\mathbf{q}\right)$ equals ${\sum }_{nm\mathbf{k}}\delta \left({\epsilon }_{\mathbf{k}}^n\right)\delta \left({\epsilon }_{\mathbf{k}+\mathbf{q}}^m\right)/N_{\mathbf{k}}$. Here, $n$ and $m$ are the indices of energy bands. ${\epsilon }_{\mathbf{k}}^n$ and ${\epsilon }_{\mathbf{k}+\mathbf{q}}^m$ represent the eigenvalues of the electronic states with respect to the Fermi energy. $N_{\mathbf{k}}$ stands for the total number of k points in the fine mesh. $\xi \left(\mathbf{q}\right)$ is normalized by $\xi \left(\mathrm{\Gamma }\right)$.

Figure 3

(a) Normalized superconducting gap distribution of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ at different temperatures. In comparison, the temperature-dependent distribution of superconducting gap in ${\mathrm{SrB}}_3{\mathrm{C}}_3$ was given. (b)–(e) The momentum-resolved EPC strength ${\lambda }_{n\mathbf{k}}$ for each electronic state $n$k on the Fermi surface. (f)–(i) Distribution of superconducting gap ${\mathrm{\Delta }}_{n\mathbf{k}}$ on the Fermi surface at 5 K.

Figure 4

(a) Top view of the $\sqrt{5}\times \sqrt{5}\times 1$ supercell for ${\mathrm{SrB}}_3{\mathrm{C}}_3$. (b) Line-type distribution of Rb atoms for ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. (c) Cluster-type distribution of Rb atoms for ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. (d) The ground-state structure for ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ in the $\sqrt{5}\times \sqrt{5}\times 1$ supercell.

Figure 5

(a) Electron DOS comparison between VCA cell and $\sqrt{5}\times \sqrt{5}\times 1$ supercell for ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$. (b) Phonon DOS comparison. In order to gain a clear view, the VCA phonon DOS is also computed in $\sqrt{5}\times \sqrt{5}\times 1$ cell, whereas Rb and Sr atoms are replaced with a virtual one containing 40% Rb and 60% Sr. The phonon DOS $F^{\mathrm{\Gamma }}\left(\omega \right)$ is obtained by the following procedure. First, we calculate the dynamical matrix at the $\mathrm{\Gamma }$ point. And then, the real-space force constants are generated by the inverse Fourier transformation of the $\mathrm{\Gamma }$-point dynamical matrix. At last, through generalized Fourier transformation, we can compute the phonon frequencies of arbitrary q points in dense grid, which are employed to construct the DOS.

Figure 6

Formation enthalpy comparison between ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ and selected reaction paths under different pressure. For clarity, the formation enthalpy of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_3{\mathrm{C}}_3$ at each pressure is set to zero.