Conduction-band structure of alkali-metal-doped C60.

Abstract

We discuss the geometries of crystalline fcc ${\mathrm{C}}_{60}$ for three cases of directional order, the hypothetical undirectional structure with space group Fm3\ifmmode\bar\else\textasciimacron\fi{}, the bidirectional structure (P${4}_{2}$/mnm) which is the ordered version of the structure observed for alkali-metal-doped compounds at room temperature, and the quadridirectional, low-temperature structure (Pa3\ifmmode\bar\else\textasciimacron\fi{}) of pure ${\mathrm{C}}_{60}$. Analytical, parameter-free expressions for the ${\mathit{t}}_{1\mathit{u}}$ wave functions are derived and used to obtain analytical conduction-band Hamiltonians for all three structures. The interactions with other subbands are included in numerical tight-binding calculations with a basis of 60 radial carbon orbitals per molecule. Ab initio density-functional [local-density approximation (LDA)] calculations are performed for unidirectional fcc ${\mathrm{C}}_{60}$ and ${\mathrm{RbC}}_{60}$ for different lattice constants. We use the linear-muffin-tin-orbitals (LMTO) method in the atomic-spheres approximation with carefully chosen interstitial spheres. The LDA bands are compared with photoemission and inverse photoemission data for ${\mathrm{C}}_{60}$. For ${\mathrm{RbC}}_{60}$ we find that the alkali-metal atom is fully ionized and that the doped electron occupies the ${\mathit{t}}_{1\mathit{u}}$ band in a rigid-band-like fashion. Tight-binding theory explains why, and indicates that this holds generally for ${\mathit{A}}_{\mathit{n}\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$${\mathrm{C}}_{60}$ with n\ensuremath{\le}3.The LDA calculation shows that, for a given structure, the conduction band scales uniformly in energy when, because of doping, the lattice constant a is changed. The energy scale behaves like W\ensuremath{\propto}d(a)exp[-d(a)/0.58 \AA{}] where d is the shortest distance between atoms belonging to different molecules. Both the LDA-LMTO and the tight-binding conduction bands are well fitted by the ${\mathit{t}}_{1\mathit{u}}$ Hamiltonians. For a=14.1 \AA{} the density of states for a conduction-band occupation of three electrons is 15, 17, and 21 electrons/mol eV) for the unidirectional, bidirectional, and quadridirectional structures, respectively. The calculated Stoner exchange parameter is less than half the inverse density of states per spin and atom, but the Coulomb self-energy for a molecular orbital is presumably larger than the ${\mathit{t}}_{1\mathit{u}}$ bandwidths, which are 0.52 eV (uni), 0.64 eV (bi), and 0.44 eV (quadri) for a=14.1 \AA{}. The LDA value (0.58 \AA{}) for the decay of the intermolecular hopping was used together with experimental data for ${\mathit{T}}_{\mathit{c}}$ vs a for ${\mathrm{K}}_{3\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Rb}}_{\mathit{x}}$${\mathrm{C}}_{60}$ compounds in the McMillan formula. The assumptions that the Coulomb interaction ${\mathrm{\ensuremath{\mu}}}^{\mathrm{*}}$, the electron-phonon interaction ${\mathit{V}}_{\mathit{e}\mathrm{\ensuremath{-}}\mathrm{ph}}$, and the average phonon frequency \ensuremath{\omega} are all independent of a were found to be inconsistent.