Advanced first-principles theory of superconductivity including both lattice vibrations and spin fluctuations: The case of FeB$_4$

We present an advanced method to study spin fluctuations in superconductors quantitatively, and entirely from first principles. This method can be generally applied to materials where electron-phonon coupling and spin fluctuations coexist. We employ it here to examine the recently synthesized superconductor iron tetraboride (FeB$_4$) with experimental $T_{\mathrm{c}}\sim 2.4$ K [H. Gou \textit{et al.}, Phys. Rev. Lett. \textbf{111}, 157002 (2013)]. We prove that FeB$_4$ is particularly prone to ferromagnetic spin fluctuations due to the presence of iron, resulting in a large Stoner interaction strength, $I=1.5$ eV, as calculated from first principles. The other important factor is its Fermi surface that consists of three separate sheets, among which two nested ellipsoids. The resulting susceptibility has a ferromagnetic peak around $\textbf{q}=0$, from which we calculated the repulsive interaction between Cooper pair electrons using the random phase approximation. Subsequently, we combined the electron-phonon interaction calculated from first principles with the spin fluctuation interaction in fully anisotropic Eliashberg theory calculations. We show that the resulting superconducting gap spectrum is conventional, yet very strongly depleted due to coupling to the spin fluctuations. The critical temperature decreases from $T_{\mathrm{c}}= 41$ K, if they are not taken into account, to $T_{\mathrm{c}}= 1.7$ K, in good agreement with the experimental value.