In this paper, the density of states and the mobility of an extra electron or hole are calculated in the atomic limit of the Hubbard model. Both the half-filled single-band and multiple-band situations are discussed. The problem is formulated in terms of the number of paths which return to the origin leaving the spin configuration unchanged. The density of states then depends on spin configuration and we have considered the random (R) (high-temperature) and antiferromagnetic (AF) arrangements. Examination of the first five nonzero moments for the simple cubic lattice indicates that the bands are narrowed by a factor of 0.745 (AF) and 0.805 (R). However, the exact bands have tails extending out to the full free-particle width for both spin arrangements. An approximate one-particle Green's function is obtained by summing all graphs with no closed loops. Such paths give a density of states that is independent of spin arrangement and is relatively flat with a sharp square-root edge at $2{\mathrm{}(z-1\mathrm{})\mathrm{}}^{\frac{1}{2}}t$. Here $z$ is the coordination number and $t$ is the nearest-neighbor hopping integral. Within this approximation, we have calculated the mobility of an extra hole and have found typical values to be ∼1 ${\mathrm{cm}}^2$/V sec so that the mobility is rather small, even though the density of states has a width of order ∼1 eV. Intra-atomic exchange is shown to give a further narrowing of the band [a factor of ${\mathrm{}\left(2\right)\mathrm{}}^{-1/2\mathrm{}}$ in the two-band large-intra-atomic-exchange example]. The effect of finite $\frac{t}{U}$ is considered, where $U$ is the intra-atomic Coulomb interaction, and is shown to have a strong effect on the band tail but relatively weak effects on the bulk of the band. Finally, we make a few remarks comparing our results with the observed mobilities in NiO and the relevance of intra-atomic exchange to the behavior of the dioxide and sesquioxide series.

• Received 5 March 1970

DOI:https://doi.org/10.1103/PhysRevB.2.1324