Orientational phases for ${\mathit{M}}_3$${\mathrm{C}}_{60}$

The mechanism of the orientational ordering of ${\mathrm{C}}_{60}$ in alkali-metal-doped fullerenes ${\mathit{M}}_3$${\mathrm{C}}_{60}$ is studied. Since the M-${\mathrm{C}}_{60}$ (M=K,Rb) interactions cause the ${\mathrm{C}}_{60}$ molecules to assume one of two standard orientations, this model is equivalent to a generalized Ising model on a fcc lattice. The Ising interactions depend on two type of energies: (1) the direct interaction, i.e., the orientationally dependent part of interactions between nearest-neighboring ${\mathrm{C}}_{60}$ molecules (each carrying charge -$3e$), and (2) the band energy of the electrons transferred from ${\mathit{M}}^{+}$ ions to the ${\mathrm{C}}_{60}^{\mathrm{-}3}$ ions. It is shown that the contribution to the pairwise interaction from the direct orientational interaction is ferromagnetic and dominantly nearest neighbor. However, contributions from the band (kinetic) energy of the conduction electrons are found to be antiferromagnetic for first- and third-nearest neighbors, ferromagnetic for second- and fourth-nearest neighbors, and negligible for further neighbors. The total first-neighbor interaction is probably antiferromagnetic. a non-negligible four-spin interaction is also obtained. The implication of these results for the orientational structure is discussed.