  We realize superconductor-insulator transitions (SIT) in mechanically
exfoliated Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (BSCCO) flakes and address
simultaneously their transport properties as well as the evolution of density
of states. Back-gating via the solid ion conductor engenders a reversible SIT
in BSCCO, as lithium ions from the substrate are electrically driven into and
out of BSCCO. Scaling analysis indicates that the SIT follows the theoretical
description of a two-dimensional quantum phase transition (2D-QPT). We further
carry out tunneling spectroscopy in graphite(G)/BSCCO heterojunctions. We
observe V-shaped gaps in the critical regime of the SIT. The density of states
in BSCCO gets symmetrically suppressed by further going into the insulating
regime. Our technique of combining solid state gating with tunneling
spectroscopy can be easily applied to the study of other two-dimensional
materials.
