Transport measurements can be used to determine the phase diagram of high temperature superconductors by detecting variations in temperature dependence of resistance in different regions of the phase diagram. While for bulk measurements several samples with varying chemical doping are used, we continuously vary carrier density in our ultra-thin two-dimensional Bi2Sr2CaCu2O[Formula: see text] device by electrostatic means and the space charge doping method. Here we concentrate on a low-disorder, high quality single unit cell thick sample. We establish the crossover to strange metal from the pseudogap and Fermi liquid phases in the normal state, close to the superconducting dome. By extrapolation we demarcate a critical doping region which is thought to correspond to a quantum phase transition at very low temperature.