Theory of superconducting ${\mathit{T}}_{\mathit{c}}$ of doped fullerenes

We develop the nonadiabatic polaron theory of superconductivity of ${\mathit{M}}_{\mathit{x}}$${\mathrm{C}}_{60}$ taking into account the polaron band narrowing and realistic electron-phonon and Coulomb interactions. We argue that the crossover from the BCS weak-coupling superconductivity to the strong-coupling polaronic and bipolaronic superconductivity occurs at the BCS coupling constant λ∼1 independent of the adiabatic ratio, and there is nothing ‘‘beyond’’ Migdal’s theorem except small polarons for any realistic electron-phonon interaction. By the use of the polaronic-type function and the ‘‘exact’’ diagonalization in the truncated Hilbert space of vibrons (‘‘phonons’’) we calculate the ground-state energy and the electron spectral density of the ${\mathrm{C}}_{60}^{\mathrm{-}}$ molecule. This allows us to describe the photoemission spectrum of ${\mathrm{C}}_{60}^{\mathrm{-}}$ in a wide energy region and determine the electron-phonon interaction. The strongest coupling is found with the high-frequency pinch ${\mathit{A}}_{\mathit{g}}$(2) mode and with the Frenkel exciton. We clarify the crucial role of high-frequency bosonic excitations in doped fullerenes which reduce the bare bandwidth and the Coulomb repulsion allowing the intermediate- and low-frequency phonons couple two small polarons in a Cooper pair. The Eliashberg-type equations are solved for low-frequency phonons. The value of the superconducting ${\mathit{T}}_{\mathit{c}}$, its pressure dependence and the isotope effect are found in remarkable agreement with the available experimental data. © 1996 The American Physical Society.