Neutron spectroscopic evidence for cluster formation and percolative superconductivity in ${\mathrm{ErBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{\mathit{x}}$

Inelastic neutron scattering has been employed to study in detail the low-energy crystal-field excitations of ${\mathrm{Er}}^{3+}$ in ${\mathrm{ErBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{\mathit{x}}$ (6<x<7). The observed energy spectra are found to be the result of a superposition of three stable states, which we interpret in terms of local regions of semiconducting and metallic character. The superconductivity is shown to result from the formation of a two-dimensional percolative network. A two-dimensional bond percolation model correctly predicts the critical oxygen contents associated with the two-plateau structure of ${\mathit{T}}_{\mathit{c}}$.