Electron-phonon coupling and superconductivity in LiB$_{1+x}$C$_{1-x}$

By means of the first-principles density-functional theory calculation and Wannier interpolation, electron-phonon coupling and superconductivity are systematically explored for boron-doped LiBC (i.e. LiB$_{1+x}$C$_{1-x}$), with $x$ between 0.1 and 0.9. Hole doping introduced by boron atoms is treated through virtual-crystal approximation. For the investigated doping concentrations, our calculations show the optimal doping concentration corresponds to 0.8. By solving the anisotropic Eliashberg equations, we find that LiB$_{1.8}$C$_{0.2}$ is a two-gap superconductor, whose superconducting transition temperature, T$_c$, may exceed the experimentally observed value of MgB$_2$. Similar to MgB$_2$, the two-dimensional bond-stretching $E_{2g}$ phonon modes along $\Gamma$-$A$ line have the largest contribution to electron-phonon coupling. More importantly, we find that the first two acoustic phonon modes $B_1$ and $A_1$ around the midpoint of $K$-$\Gamma$ line play a vital role for the rise of T$_c$ in LiB$_{1.8}$C$_{0.2}$. The origin of strong couplings in $B_1$ and $A_1$ modes can be attributed to enhanced electron-phonon coupling matrix elements and softened phonons. It is revealed that all these phonon modes couple strongly with $\sigma$-bonding electronic states.