Nanoscale phase separation in deep underdoped Bi$_{2}$Sr$_{2}$CuO$_{6+\delta}$ and Ca$_2$CuO$_2$Cl$_2$

We demonstrate that the tunneling spectra from deeply underdoped Bi$_{2}$Sr$_{2}$CuO$_{6+\delta}$ (Bi2201) and Ca$_2$CuO$_2$Cl$_2$ (CCOC) provide clear evidence for a nanoscale phase separation (NPS), which causes the gap to fill rather than close with doping. The phase separation extends from half-filling to a doping of $x\sim 0.09$. Assuming that the NPS is in the form of stripes, the nodal gap, which we model as a Coulomb gap, arises from impurity pinning of the charged stripes, and ultimately drives a metal-insulator transition.