Scaling behavior at the insulator-metal transition in ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$(${\mathrm{Ca}}_{\mathit{z}}$${\mathit{R}}_{1\mathrm{-}\mathit{z}}$)${\mathrm{Cu}}_2$${\mathrm{O}}_{8+\mathit{y}}$ where R is a rare-earth element

We report on the details of the insulator-metal transition (IMT) induced in ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$(${\mathrm{Ca}}_{\mathit{z}}$,${\mathit{R}}_{1\mathrm{-}\mathit{z}}$)${\mathrm{Cu}}_2$ ${\mathrm{O}}_{8+\mathit{y}}$ (R=Y, Gd, Nd) by ${\mathrm{Ca}}^{2+}$ doping. The resistivity in the insulating regime is analyzed using a generalized hopping approach based on the connectivity criterion. This enables us to estimate the dependence of the localization radius ${\mathit{a}}_{\mathit{H}}$ on the Ca concentration independent of dimensionality. Insulating samples with a Y content not too far from the critical concentration ${\mathit{z}}_{\mathit{C}}$=(0.43±0.02) show metallic conduction at high temperature and hopping conduction at low temperature. This shows the coexistence of delocalized and localized states separated by a disorder-induced mobility edge. Ultraviolet-photoemission-spectroscopy (UPS) spectra give evidence for both a shift in the Fermi level to lower energies and the development of new states at the Fermi level. The existence of a mobility edge together with the shift in ${\mathit{E}}_{\mathit{F}}$ upon ${\mathrm{Ca}}^{2+}$ doping shows that the transition is probably of the Anderson type. We present a schematic picture for the density of states in the vicinity of ${\mathit{E}}_{\mathit{F}}$ based on the results of spectroscopic and transport data. For this density of states we calculate the electrical resistivity using the Kubo-Greenwood formula. The results are in good qualitative agreement with the experiments. At the IMT the localization radius diverges and the metallic conductivity vanishes following a scaling law σ=${\mathrm{\sigma }}_0${1-z/${\mathit{z}}_{\mathit{C}}$ ${\mathrm{\}}}^{\mathrm{\eta }}$, with a critical exponent η=1A.